Properties

Degree 26
Conductor $ 3^{26} \cdot 7^{13} \cdot 127^{13} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 13

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 12·5-s + 13·7-s + 21·8-s + 48·10-s − 3·11-s + 21·13-s − 52·14-s − 23·16-s − 17·17-s + 5·19-s + 12·22-s − 4·23-s + 40·25-s − 84·26-s − 21·29-s − 7·31-s − 38·32-s + 68·34-s − 156·35-s + 7·37-s − 20·38-s − 252·40-s − 21·41-s − 9·43-s + 16·46-s − 23·47-s + ⋯
L(s)  = 1  − 2.82·2-s − 5.36·5-s + 4.91·7-s + 7.42·8-s + 15.1·10-s − 0.904·11-s + 5.82·13-s − 13.8·14-s − 5.75·16-s − 4.12·17-s + 1.14·19-s + 2.55·22-s − 0.834·23-s + 8·25-s − 16.4·26-s − 3.89·29-s − 1.25·31-s − 6.71·32-s + 11.6·34-s − 26.3·35-s + 1.15·37-s − 3.24·38-s − 39.8·40-s − 3.27·41-s − 1.37·43-s + 2.35·46-s − 3.35·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{26} \cdot 7^{13} \cdot 127^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{26} \cdot 7^{13} \cdot 127^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(26\)
\( N \)  =  \(3^{26} \cdot 7^{13} \cdot 127^{13}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  13
Selberg data  =  $(26,\ 3^{26} \cdot 7^{13} \cdot 127^{13} ,\ ( \ : [1/2]^{13} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \(F_p\) is a polynomial of degree 26. If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 25.
$p$$F_p$
bad3 \( 1 \)
7 \( ( 1 - T )^{13} \)
127 \( ( 1 + T )^{13} \)
good2 \( 1 + p^{2} T + p^{4} T^{2} + 43 T^{3} + 111 T^{4} + 119 p T^{5} + 491 T^{6} + 905 T^{7} + 1619 T^{8} + 167 p^{4} T^{9} + 539 p^{3} T^{10} + 819 p^{3} T^{11} + 153 p^{6} T^{12} + 1743 p^{3} T^{13} + 153 p^{7} T^{14} + 819 p^{5} T^{15} + 539 p^{6} T^{16} + 167 p^{8} T^{17} + 1619 p^{5} T^{18} + 905 p^{6} T^{19} + 491 p^{7} T^{20} + 119 p^{9} T^{21} + 111 p^{9} T^{22} + 43 p^{10} T^{23} + p^{15} T^{24} + p^{14} T^{25} + p^{13} T^{26} \)
5 \( 1 + 12 T + 104 T^{2} + 134 p T^{3} + 3638 T^{4} + 3392 p T^{5} + 70302 T^{6} + 262027 T^{7} + 178116 p T^{8} + 555764 p T^{9} + 8011459 T^{10} + 21421987 T^{11} + 53271576 T^{12} + 123398136 T^{13} + 53271576 p T^{14} + 21421987 p^{2} T^{15} + 8011459 p^{3} T^{16} + 555764 p^{5} T^{17} + 178116 p^{6} T^{18} + 262027 p^{6} T^{19} + 70302 p^{7} T^{20} + 3392 p^{9} T^{21} + 3638 p^{9} T^{22} + 134 p^{11} T^{23} + 104 p^{11} T^{24} + 12 p^{12} T^{25} + p^{13} T^{26} \)
11 \( 1 + 3 T + 83 T^{2} + 21 p T^{3} + 317 p T^{4} + 8594 T^{5} + 96020 T^{6} + 206021 T^{7} + 1938774 T^{8} + 3603777 T^{9} + 30713596 T^{10} + 50136844 T^{11} + 400155761 T^{12} + 591166452 T^{13} + 400155761 p T^{14} + 50136844 p^{2} T^{15} + 30713596 p^{3} T^{16} + 3603777 p^{4} T^{17} + 1938774 p^{5} T^{18} + 206021 p^{6} T^{19} + 96020 p^{7} T^{20} + 8594 p^{8} T^{21} + 317 p^{10} T^{22} + 21 p^{11} T^{23} + 83 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \)
13 \( 1 - 21 T + 303 T^{2} - 3206 T^{3} + 2173 p T^{4} - 211775 T^{5} + 1406812 T^{6} - 8366101 T^{7} + 45400042 T^{8} - 225933653 T^{9} + 1041869982 T^{10} - 4459348525 T^{11} + 17818659263 T^{12} - 66419150254 T^{13} + 17818659263 p T^{14} - 4459348525 p^{2} T^{15} + 1041869982 p^{3} T^{16} - 225933653 p^{4} T^{17} + 45400042 p^{5} T^{18} - 8366101 p^{6} T^{19} + 1406812 p^{7} T^{20} - 211775 p^{8} T^{21} + 2173 p^{10} T^{22} - 3206 p^{10} T^{23} + 303 p^{11} T^{24} - 21 p^{12} T^{25} + p^{13} T^{26} \)
17 \( 1 + p T + 258 T^{2} + 2538 T^{3} + 1313 p T^{4} + 152819 T^{5} + 941699 T^{6} + 274768 p T^{7} + 20527283 T^{8} + 67321211 T^{9} + 167904403 T^{10} + 10092848 T^{11} - 1775763501 T^{12} - 11811409138 T^{13} - 1775763501 p T^{14} + 10092848 p^{2} T^{15} + 167904403 p^{3} T^{16} + 67321211 p^{4} T^{17} + 20527283 p^{5} T^{18} + 274768 p^{7} T^{19} + 941699 p^{7} T^{20} + 152819 p^{8} T^{21} + 1313 p^{10} T^{22} + 2538 p^{10} T^{23} + 258 p^{11} T^{24} + p^{13} T^{25} + p^{13} T^{26} \)
19 \( 1 - 5 T + 173 T^{2} - 864 T^{3} + 14502 T^{4} - 70542 T^{5} + 782707 T^{6} - 3630922 T^{7} + 30483684 T^{8} - 132517839 T^{9} + 47863464 p T^{10} - 3646806238 T^{11} + 21484456413 T^{12} - 78131794348 T^{13} + 21484456413 p T^{14} - 3646806238 p^{2} T^{15} + 47863464 p^{4} T^{16} - 132517839 p^{4} T^{17} + 30483684 p^{5} T^{18} - 3630922 p^{6} T^{19} + 782707 p^{7} T^{20} - 70542 p^{8} T^{21} + 14502 p^{9} T^{22} - 864 p^{10} T^{23} + 173 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \)
23 \( 1 + 4 T + 8 p T^{2} + 613 T^{3} + 16621 T^{4} + 45907 T^{5} + 981152 T^{6} + 2240102 T^{7} + 1851189 p T^{8} + 80788590 T^{9} + 1448692280 T^{10} + 2339155653 T^{11} + 40151933279 T^{12} + 57649846102 T^{13} + 40151933279 p T^{14} + 2339155653 p^{2} T^{15} + 1448692280 p^{3} T^{16} + 80788590 p^{4} T^{17} + 1851189 p^{6} T^{18} + 2240102 p^{6} T^{19} + 981152 p^{7} T^{20} + 45907 p^{8} T^{21} + 16621 p^{9} T^{22} + 613 p^{10} T^{23} + 8 p^{12} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \)
29 \( 1 + 21 T + 421 T^{2} + 5324 T^{3} + 64636 T^{4} + 613764 T^{5} + 5683567 T^{6} + 44367699 T^{7} + 342010383 T^{8} + 2302116496 T^{9} + 15447960508 T^{10} + 92186196428 T^{11} + 551990408672 T^{12} + 2963927041432 T^{13} + 551990408672 p T^{14} + 92186196428 p^{2} T^{15} + 15447960508 p^{3} T^{16} + 2302116496 p^{4} T^{17} + 342010383 p^{5} T^{18} + 44367699 p^{6} T^{19} + 5683567 p^{7} T^{20} + 613764 p^{8} T^{21} + 64636 p^{9} T^{22} + 5324 p^{10} T^{23} + 421 p^{11} T^{24} + 21 p^{12} T^{25} + p^{13} T^{26} \)
31 \( 1 + 7 T + 7 p T^{2} + 1437 T^{3} + 23361 T^{4} + 141394 T^{5} + 1651220 T^{6} + 9026082 T^{7} + 86549696 T^{8} + 428736095 T^{9} + 3637767504 T^{10} + 16521213633 T^{11} + 129439998267 T^{12} + 546785963664 T^{13} + 129439998267 p T^{14} + 16521213633 p^{2} T^{15} + 3637767504 p^{3} T^{16} + 428736095 p^{4} T^{17} + 86549696 p^{5} T^{18} + 9026082 p^{6} T^{19} + 1651220 p^{7} T^{20} + 141394 p^{8} T^{21} + 23361 p^{9} T^{22} + 1437 p^{10} T^{23} + 7 p^{12} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \)
37 \( 1 - 7 T + 349 T^{2} - 2305 T^{3} + 57994 T^{4} - 369015 T^{5} + 6147489 T^{6} - 37952961 T^{7} + 468210685 T^{8} - 2784936853 T^{9} + 27265286966 T^{10} - 153098638011 T^{11} + 1255909686356 T^{12} - 6449571361272 T^{13} + 1255909686356 p T^{14} - 153098638011 p^{2} T^{15} + 27265286966 p^{3} T^{16} - 2784936853 p^{4} T^{17} + 468210685 p^{5} T^{18} - 37952961 p^{6} T^{19} + 6147489 p^{7} T^{20} - 369015 p^{8} T^{21} + 57994 p^{9} T^{22} - 2305 p^{10} T^{23} + 349 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \)
41 \( 1 + 21 T + 616 T^{2} + 9402 T^{3} + 160488 T^{4} + 1938936 T^{5} + 24485336 T^{6} + 245110356 T^{7} + 2497863778 T^{8} + 21278259882 T^{9} + 182459787904 T^{10} + 1342887611296 T^{11} + 9902669942613 T^{12} + 63386892075414 T^{13} + 9902669942613 p T^{14} + 1342887611296 p^{2} T^{15} + 182459787904 p^{3} T^{16} + 21278259882 p^{4} T^{17} + 2497863778 p^{5} T^{18} + 245110356 p^{6} T^{19} + 24485336 p^{7} T^{20} + 1938936 p^{8} T^{21} + 160488 p^{9} T^{22} + 9402 p^{10} T^{23} + 616 p^{11} T^{24} + 21 p^{12} T^{25} + p^{13} T^{26} \)
43 \( 1 + 9 T + 363 T^{2} + 2829 T^{3} + 62991 T^{4} + 434130 T^{5} + 7053472 T^{6} + 43885133 T^{7} + 580184802 T^{8} + 3304680495 T^{9} + 37503098992 T^{10} + 196157482974 T^{11} + 1966417783053 T^{12} + 9376737802572 T^{13} + 1966417783053 p T^{14} + 196157482974 p^{2} T^{15} + 37503098992 p^{3} T^{16} + 3304680495 p^{4} T^{17} + 580184802 p^{5} T^{18} + 43885133 p^{6} T^{19} + 7053472 p^{7} T^{20} + 434130 p^{8} T^{21} + 62991 p^{9} T^{22} + 2829 p^{10} T^{23} + 363 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \)
47 \( 1 + 23 T + 607 T^{2} + 9855 T^{3} + 158534 T^{4} + 2026095 T^{5} + 24946377 T^{6} + 265384987 T^{7} + 2704629542 T^{8} + 24695815357 T^{9} + 215886050220 T^{10} + 1720118801318 T^{11} + 13126247577063 T^{12} + 91942347287866 T^{13} + 13126247577063 p T^{14} + 1720118801318 p^{2} T^{15} + 215886050220 p^{3} T^{16} + 24695815357 p^{4} T^{17} + 2704629542 p^{5} T^{18} + 265384987 p^{6} T^{19} + 24946377 p^{7} T^{20} + 2026095 p^{8} T^{21} + 158534 p^{9} T^{22} + 9855 p^{10} T^{23} + 607 p^{11} T^{24} + 23 p^{12} T^{25} + p^{13} T^{26} \)
53 \( 1 + 31 T + 936 T^{2} + 18071 T^{3} + 328569 T^{4} + 4749105 T^{5} + 64916194 T^{6} + 760070261 T^{7} + 8473173808 T^{8} + 83734170303 T^{9} + 791809806079 T^{10} + 6750789985461 T^{11} + 55228222405121 T^{12} + 410273915363592 T^{13} + 55228222405121 p T^{14} + 6750789985461 p^{2} T^{15} + 791809806079 p^{3} T^{16} + 83734170303 p^{4} T^{17} + 8473173808 p^{5} T^{18} + 760070261 p^{6} T^{19} + 64916194 p^{7} T^{20} + 4749105 p^{8} T^{21} + 328569 p^{9} T^{22} + 18071 p^{10} T^{23} + 936 p^{11} T^{24} + 31 p^{12} T^{25} + p^{13} T^{26} \)
59 \( 1 + 28 T + 748 T^{2} + 12353 T^{3} + 200007 T^{4} + 2482731 T^{5} + 31037042 T^{6} + 320236588 T^{7} + 3396643521 T^{8} + 30749512300 T^{9} + 290422351830 T^{10} + 2376472711159 T^{11} + 20434393717531 T^{12} + 152974888166578 T^{13} + 20434393717531 p T^{14} + 2376472711159 p^{2} T^{15} + 290422351830 p^{3} T^{16} + 30749512300 p^{4} T^{17} + 3396643521 p^{5} T^{18} + 320236588 p^{6} T^{19} + 31037042 p^{7} T^{20} + 2482731 p^{8} T^{21} + 200007 p^{9} T^{22} + 12353 p^{10} T^{23} + 748 p^{11} T^{24} + 28 p^{12} T^{25} + p^{13} T^{26} \)
61 \( 1 - 29 T + 1023 T^{2} - 20439 T^{3} + 425921 T^{4} - 6588276 T^{5} + 102071192 T^{6} - 1292237308 T^{7} + 16152643052 T^{8} - 172446855471 T^{9} + 1807344944080 T^{10} - 16534828328737 T^{11} + 148161174495311 T^{12} - 1169622992942008 T^{13} + 148161174495311 p T^{14} - 16534828328737 p^{2} T^{15} + 1807344944080 p^{3} T^{16} - 172446855471 p^{4} T^{17} + 16152643052 p^{5} T^{18} - 1292237308 p^{6} T^{19} + 102071192 p^{7} T^{20} - 6588276 p^{8} T^{21} + 425921 p^{9} T^{22} - 20439 p^{10} T^{23} + 1023 p^{11} T^{24} - 29 p^{12} T^{25} + p^{13} T^{26} \)
67 \( 1 + 18 T + 515 T^{2} + 7308 T^{3} + 124920 T^{4} + 1475761 T^{5} + 19125783 T^{6} + 196479200 T^{7} + 2123208520 T^{8} + 19682935334 T^{9} + 187612591758 T^{10} + 1616331925468 T^{11} + 14149667857895 T^{12} + 115112160627838 T^{13} + 14149667857895 p T^{14} + 1616331925468 p^{2} T^{15} + 187612591758 p^{3} T^{16} + 19682935334 p^{4} T^{17} + 2123208520 p^{5} T^{18} + 196479200 p^{6} T^{19} + 19125783 p^{7} T^{20} + 1475761 p^{8} T^{21} + 124920 p^{9} T^{22} + 7308 p^{10} T^{23} + 515 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \)
71 \( 1 + 10 T + 556 T^{2} + 4573 T^{3} + 149491 T^{4} + 1052465 T^{5} + 26282634 T^{6} + 161622081 T^{7} + 3414944265 T^{8} + 18615640006 T^{9} + 350734762938 T^{10} + 1720351297082 T^{11} + 29653858976227 T^{12} + 132764914433646 T^{13} + 29653858976227 p T^{14} + 1720351297082 p^{2} T^{15} + 350734762938 p^{3} T^{16} + 18615640006 p^{4} T^{17} + 3414944265 p^{5} T^{18} + 161622081 p^{6} T^{19} + 26282634 p^{7} T^{20} + 1052465 p^{8} T^{21} + 149491 p^{9} T^{22} + 4573 p^{10} T^{23} + 556 p^{11} T^{24} + 10 p^{12} T^{25} + p^{13} T^{26} \)
73 \( 1 - 24 T + 870 T^{2} - 15387 T^{3} + 325681 T^{4} - 4669441 T^{5} + 74019844 T^{6} - 905055656 T^{7} + 11773114747 T^{8} - 126039249920 T^{9} + 1404528656478 T^{10} - 13325817517321 T^{11} + 130094309435051 T^{12} - 1097731775448310 T^{13} + 130094309435051 p T^{14} - 13325817517321 p^{2} T^{15} + 1404528656478 p^{3} T^{16} - 126039249920 p^{4} T^{17} + 11773114747 p^{5} T^{18} - 905055656 p^{6} T^{19} + 74019844 p^{7} T^{20} - 4669441 p^{8} T^{21} + 325681 p^{9} T^{22} - 15387 p^{10} T^{23} + 870 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \)
79 \( 1 + 28 T + 1025 T^{2} + 19014 T^{3} + 398721 T^{4} + 5420565 T^{5} + 81531672 T^{6} + 832892678 T^{7} + 9725170712 T^{8} + 72303138705 T^{9} + 699331712541 T^{10} + 3412737965740 T^{11} + 35417479052632 T^{12} + 139160460017484 T^{13} + 35417479052632 p T^{14} + 3412737965740 p^{2} T^{15} + 699331712541 p^{3} T^{16} + 72303138705 p^{4} T^{17} + 9725170712 p^{5} T^{18} + 832892678 p^{6} T^{19} + 81531672 p^{7} T^{20} + 5420565 p^{8} T^{21} + 398721 p^{9} T^{22} + 19014 p^{10} T^{23} + 1025 p^{11} T^{24} + 28 p^{12} T^{25} + p^{13} T^{26} \)
83 \( 1 + 26 T + 912 T^{2} + 16870 T^{3} + 351403 T^{4} + 5167744 T^{5} + 81355420 T^{6} + 1007182561 T^{7} + 13151871421 T^{8} + 142231819988 T^{9} + 1621508875722 T^{10} + 15730315041009 T^{11} + 161523128267333 T^{12} + 1428718611245444 T^{13} + 161523128267333 p T^{14} + 15730315041009 p^{2} T^{15} + 1621508875722 p^{3} T^{16} + 142231819988 p^{4} T^{17} + 13151871421 p^{5} T^{18} + 1007182561 p^{6} T^{19} + 81355420 p^{7} T^{20} + 5167744 p^{8} T^{21} + 351403 p^{9} T^{22} + 16870 p^{10} T^{23} + 912 p^{11} T^{24} + 26 p^{12} T^{25} + p^{13} T^{26} \)
89 \( 1 + 44 T + 1789 T^{2} + 48025 T^{3} + 1179457 T^{4} + 23460109 T^{5} + 431592848 T^{6} + 6880773166 T^{7} + 102377806578 T^{8} + 1360607322250 T^{9} + 16975488400752 T^{10} + 191986074688377 T^{11} + 2045012097441543 T^{12} + 19868978401857274 T^{13} + 2045012097441543 p T^{14} + 191986074688377 p^{2} T^{15} + 16975488400752 p^{3} T^{16} + 1360607322250 p^{4} T^{17} + 102377806578 p^{5} T^{18} + 6880773166 p^{6} T^{19} + 431592848 p^{7} T^{20} + 23460109 p^{8} T^{21} + 1179457 p^{9} T^{22} + 48025 p^{10} T^{23} + 1789 p^{11} T^{24} + 44 p^{12} T^{25} + p^{13} T^{26} \)
97 \( 1 - 17 T + 998 T^{2} - 14430 T^{3} + 471225 T^{4} - 5972993 T^{5} + 141228807 T^{6} - 1591368836 T^{7} + 30095354665 T^{8} - 303043084483 T^{9} + 4820083155707 T^{10} - 43368648096954 T^{11} + 597070384172585 T^{12} - 4777592162764534 T^{13} + 597070384172585 p T^{14} - 43368648096954 p^{2} T^{15} + 4820083155707 p^{3} T^{16} - 303043084483 p^{4} T^{17} + 30095354665 p^{5} T^{18} - 1591368836 p^{6} T^{19} + 141228807 p^{7} T^{20} - 5972993 p^{8} T^{21} + 471225 p^{9} T^{22} - 14430 p^{10} T^{23} + 998 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.47929709767864894918569567434, −2.44600041865091069018085035753, −2.44096605029810718843551568927, −2.39682580938357416322277817963, −2.11916747369891743343312801012, −1.99665350770640841284099339829, −1.98287533180890307041429277163, −1.96830099579191750490943233214, −1.94745489021789217979004090629, −1.93757126535635117660649444368, −1.80312502875150527816709895254, −1.77340315613627045001134263824, −1.76761288060562782071768442327, −1.51399440962788534261189351418, −1.36184883427356093035698038169, −1.35138407809896135602577995870, −1.26524132231775778186577920827, −1.24908626536862198568871970716, −1.22167274262807304499687207290, −1.18339700259551294861121555385, −1.13595834172652689410151164670, −1.03816421937151758153099926746, −0.991614986287796923906064892692, −0.897165543011092017387848736399, −0.872644942956393490708974784664, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.872644942956393490708974784664, 0.897165543011092017387848736399, 0.991614986287796923906064892692, 1.03816421937151758153099926746, 1.13595834172652689410151164670, 1.18339700259551294861121555385, 1.22167274262807304499687207290, 1.24908626536862198568871970716, 1.26524132231775778186577920827, 1.35138407809896135602577995870, 1.36184883427356093035698038169, 1.51399440962788534261189351418, 1.76761288060562782071768442327, 1.77340315613627045001134263824, 1.80312502875150527816709895254, 1.93757126535635117660649444368, 1.94745489021789217979004090629, 1.96830099579191750490943233214, 1.98287533180890307041429277163, 1.99665350770640841284099339829, 2.11916747369891743343312801012, 2.39682580938357416322277817963, 2.44096605029810718843551568927, 2.44600041865091069018085035753, 2.47929709767864894918569567434

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.