Properties

Degree 26
Conductor $ 2^{52} \cdot 3^{13} \cdot 167^{13} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 13·3-s + 2·5-s − 7-s + 91·9-s − 11·11-s + 12·13-s − 26·15-s + 15·17-s − 14·19-s + 13·21-s − 9·23-s − 12·25-s − 455·27-s − 3·29-s + 17·31-s + 143·33-s − 2·35-s + 16·37-s − 156·39-s + 12·41-s − 20·43-s + 182·45-s + 6·47-s − 32·49-s − 195·51-s − 12·53-s − 22·55-s + ⋯
L(s)  = 1  − 7.50·3-s + 0.894·5-s − 0.377·7-s + 91/3·9-s − 3.31·11-s + 3.32·13-s − 6.71·15-s + 3.63·17-s − 3.21·19-s + 2.83·21-s − 1.87·23-s − 2.39·25-s − 87.5·27-s − 0.557·29-s + 3.05·31-s + 24.8·33-s − 0.338·35-s + 2.63·37-s − 24.9·39-s + 1.87·41-s − 3.04·43-s + 27.1·45-s + 0.875·47-s − 4.57·49-s − 27.3·51-s − 1.64·53-s − 2.96·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{52} \cdot 3^{13} \cdot 167^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{52} \cdot 3^{13} \cdot 167^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(26\)
\( N \)  =  \(2^{52} \cdot 3^{13} \cdot 167^{13}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(26,\ 2^{52} \cdot 3^{13} \cdot 167^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )$
$L(1)$  $\approx$  $0.3787299817$
$L(\frac12)$  $\approx$  $0.3787299817$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \(F_p\) is a polynomial of degree 26. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 25.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 + T )^{13} \)
167 \( ( 1 - T )^{13} \)
good5 \( 1 - 2 T + 16 T^{2} - 21 T^{3} + 156 T^{4} - 229 T^{5} + 1338 T^{6} - 1817 T^{7} + 8691 T^{8} - 98 p^{3} T^{9} + 53818 T^{10} - 3128 p^{2} T^{11} + 288836 T^{12} - 394106 T^{13} + 288836 p T^{14} - 3128 p^{4} T^{15} + 53818 p^{3} T^{16} - 98 p^{7} T^{17} + 8691 p^{5} T^{18} - 1817 p^{6} T^{19} + 1338 p^{7} T^{20} - 229 p^{8} T^{21} + 156 p^{9} T^{22} - 21 p^{10} T^{23} + 16 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \)
7 \( 1 + T + 33 T^{2} + 3 T^{3} + 503 T^{4} - 354 T^{5} + 5636 T^{6} - 5895 T^{7} + 58851 T^{8} - 56585 T^{9} + 539397 T^{10} - 532940 T^{11} + 4140879 T^{12} - 4388332 T^{13} + 4140879 p T^{14} - 532940 p^{2} T^{15} + 539397 p^{3} T^{16} - 56585 p^{4} T^{17} + 58851 p^{5} T^{18} - 5895 p^{6} T^{19} + 5636 p^{7} T^{20} - 354 p^{8} T^{21} + 503 p^{9} T^{22} + 3 p^{10} T^{23} + 33 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \)
11 \( 1 + p T + 114 T^{2} + 68 p T^{3} + 4668 T^{4} + 22829 T^{5} + 109910 T^{6} + 452044 T^{7} + 1901898 T^{8} + 7177201 T^{9} + 28190517 T^{10} + 830296 p^{2} T^{11} + 367612018 T^{12} + 1210062606 T^{13} + 367612018 p T^{14} + 830296 p^{4} T^{15} + 28190517 p^{3} T^{16} + 7177201 p^{4} T^{17} + 1901898 p^{5} T^{18} + 452044 p^{6} T^{19} + 109910 p^{7} T^{20} + 22829 p^{8} T^{21} + 4668 p^{9} T^{22} + 68 p^{11} T^{23} + 114 p^{11} T^{24} + p^{13} T^{25} + p^{13} T^{26} \)
13 \( 1 - 12 T + 108 T^{2} - 768 T^{3} + 29 p^{2} T^{4} - 27045 T^{5} + 139081 T^{6} - 660978 T^{7} + 2991458 T^{8} - 12740044 T^{9} + 52566005 T^{10} - 207026958 T^{11} + 790583670 T^{12} - 2888715078 T^{13} + 790583670 p T^{14} - 207026958 p^{2} T^{15} + 52566005 p^{3} T^{16} - 12740044 p^{4} T^{17} + 2991458 p^{5} T^{18} - 660978 p^{6} T^{19} + 139081 p^{7} T^{20} - 27045 p^{8} T^{21} + 29 p^{11} T^{22} - 768 p^{10} T^{23} + 108 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \)
17 \( 1 - 15 T + 167 T^{2} - 1346 T^{3} + 9671 T^{4} - 3453 p T^{5} + 325131 T^{6} - 1572070 T^{7} + 7060135 T^{8} - 28216201 T^{9} + 107281433 T^{10} - 374119928 T^{11} + 1367940006 T^{12} - 5174549350 T^{13} + 1367940006 p T^{14} - 374119928 p^{2} T^{15} + 107281433 p^{3} T^{16} - 28216201 p^{4} T^{17} + 7060135 p^{5} T^{18} - 1572070 p^{6} T^{19} + 325131 p^{7} T^{20} - 3453 p^{9} T^{21} + 9671 p^{9} T^{22} - 1346 p^{10} T^{23} + 167 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \)
19 \( 1 + 14 T + 222 T^{2} + 2068 T^{3} + 19253 T^{4} + 135689 T^{5} + 932679 T^{6} + 5301112 T^{7} + 29447836 T^{8} + 7453730 p T^{9} + 680098931 T^{10} + 2942567796 T^{11} + 13246753430 T^{12} + 55845390246 T^{13} + 13246753430 p T^{14} + 2942567796 p^{2} T^{15} + 680098931 p^{3} T^{16} + 7453730 p^{5} T^{17} + 29447836 p^{5} T^{18} + 5301112 p^{6} T^{19} + 932679 p^{7} T^{20} + 135689 p^{8} T^{21} + 19253 p^{9} T^{22} + 2068 p^{10} T^{23} + 222 p^{11} T^{24} + 14 p^{12} T^{25} + p^{13} T^{26} \)
23 \( 1 + 9 T + 90 T^{2} + 376 T^{3} + 2002 T^{4} + 2087 T^{5} + 13880 T^{6} - 88608 T^{7} + 177222 T^{8} - 623197 T^{9} + 21435133 T^{10} + 90849960 T^{11} + 949979326 T^{12} + 3321483482 T^{13} + 949979326 p T^{14} + 90849960 p^{2} T^{15} + 21435133 p^{3} T^{16} - 623197 p^{4} T^{17} + 177222 p^{5} T^{18} - 88608 p^{6} T^{19} + 13880 p^{7} T^{20} + 2087 p^{8} T^{21} + 2002 p^{9} T^{22} + 376 p^{10} T^{23} + 90 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \)
29 \( 1 + 3 T + 226 T^{2} + 744 T^{3} + 26600 T^{4} + 88729 T^{5} + 2101656 T^{6} + 6843616 T^{7} + 122885736 T^{8} + 13111453 p T^{9} + 5575564329 T^{10} + 16029028280 T^{11} + 201175820032 T^{12} + 525255655926 T^{13} + 201175820032 p T^{14} + 16029028280 p^{2} T^{15} + 5575564329 p^{3} T^{16} + 13111453 p^{5} T^{17} + 122885736 p^{5} T^{18} + 6843616 p^{6} T^{19} + 2101656 p^{7} T^{20} + 88729 p^{8} T^{21} + 26600 p^{9} T^{22} + 744 p^{10} T^{23} + 226 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \)
31 \( 1 - 17 T + 354 T^{2} - 4502 T^{3} + 57000 T^{4} - 582684 T^{5} + 5671184 T^{6} - 48655573 T^{7} + 394190897 T^{8} - 2915699897 T^{9} + 20375864258 T^{10} - 132029977589 T^{11} + 809685079210 T^{12} - 4632244947956 T^{13} + 809685079210 p T^{14} - 132029977589 p^{2} T^{15} + 20375864258 p^{3} T^{16} - 2915699897 p^{4} T^{17} + 394190897 p^{5} T^{18} - 48655573 p^{6} T^{19} + 5671184 p^{7} T^{20} - 582684 p^{8} T^{21} + 57000 p^{9} T^{22} - 4502 p^{10} T^{23} + 354 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \)
37 \( 1 - 16 T + 384 T^{2} - 4711 T^{3} + 66546 T^{4} - 675545 T^{5} + 7204628 T^{6} - 63112033 T^{7} + 556641667 T^{8} - 116327432 p T^{9} + 32771180488 T^{10} - 226104266132 T^{11} + 1519031839710 T^{12} - 9379771706774 T^{13} + 1519031839710 p T^{14} - 226104266132 p^{2} T^{15} + 32771180488 p^{3} T^{16} - 116327432 p^{5} T^{17} + 556641667 p^{5} T^{18} - 63112033 p^{6} T^{19} + 7204628 p^{7} T^{20} - 675545 p^{8} T^{21} + 66546 p^{9} T^{22} - 4711 p^{10} T^{23} + 384 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \)
41 \( 1 - 12 T + 323 T^{2} - 2549 T^{3} + 41774 T^{4} - 231699 T^{5} + 3266838 T^{6} - 13225111 T^{7} + 194237287 T^{8} - 576726116 T^{9} + 9649549285 T^{10} - 20359747964 T^{11} + 418471414556 T^{12} - 720964412298 T^{13} + 418471414556 p T^{14} - 20359747964 p^{2} T^{15} + 9649549285 p^{3} T^{16} - 576726116 p^{4} T^{17} + 194237287 p^{5} T^{18} - 13225111 p^{6} T^{19} + 3266838 p^{7} T^{20} - 231699 p^{8} T^{21} + 41774 p^{9} T^{22} - 2549 p^{10} T^{23} + 323 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \)
43 \( 1 + 20 T + 447 T^{2} + 6085 T^{3} + 81758 T^{4} + 883477 T^{5} + 212472 p T^{6} + 83940113 T^{7} + 732178595 T^{8} + 5916935844 T^{9} + 45657111985 T^{10} + 332352060170 T^{11} + 2333446642614 T^{12} + 15546056453590 T^{13} + 2333446642614 p T^{14} + 332352060170 p^{2} T^{15} + 45657111985 p^{3} T^{16} + 5916935844 p^{4} T^{17} + 732178595 p^{5} T^{18} + 83940113 p^{6} T^{19} + 212472 p^{8} T^{20} + 883477 p^{8} T^{21} + 81758 p^{9} T^{22} + 6085 p^{10} T^{23} + 447 p^{11} T^{24} + 20 p^{12} T^{25} + p^{13} T^{26} \)
47 \( 1 - 6 T + 260 T^{2} - 1150 T^{3} + 34866 T^{4} - 115419 T^{5} + 3258908 T^{6} - 8213299 T^{7} + 240130821 T^{8} - 465844244 T^{9} + 14793482024 T^{10} - 23168213451 T^{11} + 789796106976 T^{12} - 1097130074574 T^{13} + 789796106976 p T^{14} - 23168213451 p^{2} T^{15} + 14793482024 p^{3} T^{16} - 465844244 p^{4} T^{17} + 240130821 p^{5} T^{18} - 8213299 p^{6} T^{19} + 3258908 p^{7} T^{20} - 115419 p^{8} T^{21} + 34866 p^{9} T^{22} - 1150 p^{10} T^{23} + 260 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \)
53 \( 1 + 12 T + 546 T^{2} + 5632 T^{3} + 139434 T^{4} + 1252903 T^{5} + 22234644 T^{6} + 175977017 T^{7} + 2496799673 T^{8} + 17567770072 T^{9} + 211114344916 T^{10} + 1329580708237 T^{11} + 13992947725186 T^{12} + 79044335106166 T^{13} + 13992947725186 p T^{14} + 1329580708237 p^{2} T^{15} + 211114344916 p^{3} T^{16} + 17567770072 p^{4} T^{17} + 2496799673 p^{5} T^{18} + 175977017 p^{6} T^{19} + 22234644 p^{7} T^{20} + 1252903 p^{8} T^{21} + 139434 p^{9} T^{22} + 5632 p^{10} T^{23} + 546 p^{11} T^{24} + 12 p^{12} T^{25} + p^{13} T^{26} \)
59 \( 1 + 14 T + 580 T^{2} + 7718 T^{3} + 2838 p T^{4} + 2023305 T^{5} + 31301076 T^{6} + 336139055 T^{7} + 4167954839 T^{8} + 39566719048 T^{9} + 414256794384 T^{10} + 3477881140059 T^{11} + 31532600144230 T^{12} + 233818570878866 T^{13} + 31532600144230 p T^{14} + 3477881140059 p^{2} T^{15} + 414256794384 p^{3} T^{16} + 39566719048 p^{4} T^{17} + 4167954839 p^{5} T^{18} + 336139055 p^{6} T^{19} + 31301076 p^{7} T^{20} + 2023305 p^{8} T^{21} + 2838 p^{10} T^{22} + 7718 p^{10} T^{23} + 580 p^{11} T^{24} + 14 p^{12} T^{25} + p^{13} T^{26} \)
61 \( 1 - 24 T + 706 T^{2} - 12276 T^{3} + 214433 T^{4} - 2938037 T^{5} + 38645919 T^{6} - 439777380 T^{7} + 4761251434 T^{8} - 46772112756 T^{9} + 438402378815 T^{10} - 3835108282184 T^{11} + 32216109760932 T^{12} - 256351733098142 T^{13} + 32216109760932 p T^{14} - 3835108282184 p^{2} T^{15} + 438402378815 p^{3} T^{16} - 46772112756 p^{4} T^{17} + 4761251434 p^{5} T^{18} - 439777380 p^{6} T^{19} + 38645919 p^{7} T^{20} - 2938037 p^{8} T^{21} + 214433 p^{9} T^{22} - 12276 p^{10} T^{23} + 706 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \)
67 \( 1 + 3 T + 355 T^{2} + 1048 T^{3} + 65889 T^{4} + 183700 T^{5} + 8652330 T^{6} + 21924177 T^{7} + 907926505 T^{8} + 2115795481 T^{9} + 80535141351 T^{10} + 178166589169 T^{11} + 6181969561833 T^{12} + 193396826556 p T^{13} + 6181969561833 p T^{14} + 178166589169 p^{2} T^{15} + 80535141351 p^{3} T^{16} + 2115795481 p^{4} T^{17} + 907926505 p^{5} T^{18} + 21924177 p^{6} T^{19} + 8652330 p^{7} T^{20} + 183700 p^{8} T^{21} + 65889 p^{9} T^{22} + 1048 p^{10} T^{23} + 355 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \)
71 \( 1 + 17 T + 649 T^{2} + 9605 T^{3} + 199299 T^{4} + 2563593 T^{5} + 38342131 T^{6} + 432635821 T^{7} + 5216366827 T^{8} + 52468153659 T^{9} + 542623256987 T^{10} + 4955284901374 T^{11} + 45797136160416 T^{12} + 384369572541270 T^{13} + 45797136160416 p T^{14} + 4955284901374 p^{2} T^{15} + 542623256987 p^{3} T^{16} + 52468153659 p^{4} T^{17} + 5216366827 p^{5} T^{18} + 432635821 p^{6} T^{19} + 38342131 p^{7} T^{20} + 2563593 p^{8} T^{21} + 199299 p^{9} T^{22} + 9605 p^{10} T^{23} + 649 p^{11} T^{24} + 17 p^{12} T^{25} + p^{13} T^{26} \)
73 \( 1 - 34 T + 1160 T^{2} - 25848 T^{3} + 540301 T^{4} - 9197595 T^{5} + 146223451 T^{6} - 2032969044 T^{7} + 26498553004 T^{8} - 311457231682 T^{9} + 3445866716883 T^{10} - 34878150258260 T^{11} + 333165209029664 T^{12} - 2928902375880834 T^{13} + 333165209029664 p T^{14} - 34878150258260 p^{2} T^{15} + 3445866716883 p^{3} T^{16} - 311457231682 p^{4} T^{17} + 26498553004 p^{5} T^{18} - 2032969044 p^{6} T^{19} + 146223451 p^{7} T^{20} - 9197595 p^{8} T^{21} + 540301 p^{9} T^{22} - 25848 p^{10} T^{23} + 1160 p^{11} T^{24} - 34 p^{12} T^{25} + p^{13} T^{26} \)
79 \( 1 + 10 T + 727 T^{2} + 6839 T^{3} + 250318 T^{4} + 2246253 T^{5} + 54783194 T^{6} + 472895673 T^{7} + 8629118749 T^{8} + 71522751906 T^{9} + 1046955450079 T^{10} + 8206987443824 T^{11} + 101626477677844 T^{12} + 732512143274958 T^{13} + 101626477677844 p T^{14} + 8206987443824 p^{2} T^{15} + 1046955450079 p^{3} T^{16} + 71522751906 p^{4} T^{17} + 8629118749 p^{5} T^{18} + 472895673 p^{6} T^{19} + 54783194 p^{7} T^{20} + 2246253 p^{8} T^{21} + 250318 p^{9} T^{22} + 6839 p^{10} T^{23} + 727 p^{11} T^{24} + 10 p^{12} T^{25} + p^{13} T^{26} \)
83 \( 1 + 44 T + 1370 T^{2} + 31288 T^{3} + 593606 T^{4} + 9457979 T^{5} + 131609258 T^{6} + 1600854901 T^{7} + 17366694499 T^{8} + 167988308462 T^{9} + 1482913205406 T^{10} + 12175793267931 T^{11} + 99368916657416 T^{12} + 857290020851094 T^{13} + 99368916657416 p T^{14} + 12175793267931 p^{2} T^{15} + 1482913205406 p^{3} T^{16} + 167988308462 p^{4} T^{17} + 17366694499 p^{5} T^{18} + 1600854901 p^{6} T^{19} + 131609258 p^{7} T^{20} + 9457979 p^{8} T^{21} + 593606 p^{9} T^{22} + 31288 p^{10} T^{23} + 1370 p^{11} T^{24} + 44 p^{12} T^{25} + p^{13} T^{26} \)
89 \( 1 - 25 T + 1039 T^{2} - 18532 T^{3} + 444053 T^{4} - 6196410 T^{5} + 110229192 T^{6} - 1267330941 T^{7} + 18562991951 T^{8} - 182639744007 T^{9} + 2348123360619 T^{10} - 20519519261863 T^{11} + 242328807802313 T^{12} - 1950518612453148 T^{13} + 242328807802313 p T^{14} - 20519519261863 p^{2} T^{15} + 2348123360619 p^{3} T^{16} - 182639744007 p^{4} T^{17} + 18562991951 p^{5} T^{18} - 1267330941 p^{6} T^{19} + 110229192 p^{7} T^{20} - 6196410 p^{8} T^{21} + 444053 p^{9} T^{22} - 18532 p^{10} T^{23} + 1039 p^{11} T^{24} - 25 p^{12} T^{25} + p^{13} T^{26} \)
97 \( 1 - 38 T + 1091 T^{2} - 23273 T^{3} + 444655 T^{4} - 7315883 T^{5} + 111741834 T^{6} - 1547021699 T^{7} + 20265183049 T^{8} - 246248874784 T^{9} + 2866188972669 T^{10} - 31400133858964 T^{11} + 331888842256893 T^{12} - 3323054133939038 T^{13} + 331888842256893 p T^{14} - 31400133858964 p^{2} T^{15} + 2866188972669 p^{3} T^{16} - 246248874784 p^{4} T^{17} + 20265183049 p^{5} T^{18} - 1547021699 p^{6} T^{19} + 111741834 p^{7} T^{20} - 7315883 p^{8} T^{21} + 444655 p^{9} T^{22} - 23273 p^{10} T^{23} + 1091 p^{11} T^{24} - 38 p^{12} T^{25} + p^{13} T^{26} \)
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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

−1.98420257226293536166453664776, −1.86359497112076174620157233746, −1.75212507987232970629288852219, −1.68943499919139246665170233634, −1.66952229967608476220811418001, −1.66538724268454926844756238908, −1.66262567242642535821645263438, −1.64352990704299605335983856968, −1.60995105628986370670513022320, −1.55543027547813372653103583052, −1.46813786678845969533585755582, −1.21320863789813250333520983365, −1.07434519172689328599194454799, −0.974214997029111080202106368267, −0.888596868404818975717313597860, −0.861476369774250741556415320596, −0.822973061741726542234016974706, −0.76361153578716861587058377981, −0.53494854254086693540266079804, −0.49731862659161196021048131806, −0.46108285108728015078980882500, −0.40123076677850108841055923813, −0.34870196825135084370196334876, −0.23363783267227431755367386218, −0.07605181808885165553417481566, 0.07605181808885165553417481566, 0.23363783267227431755367386218, 0.34870196825135084370196334876, 0.40123076677850108841055923813, 0.46108285108728015078980882500, 0.49731862659161196021048131806, 0.53494854254086693540266079804, 0.76361153578716861587058377981, 0.822973061741726542234016974706, 0.861476369774250741556415320596, 0.888596868404818975717313597860, 0.974214997029111080202106368267, 1.07434519172689328599194454799, 1.21320863789813250333520983365, 1.46813786678845969533585755582, 1.55543027547813372653103583052, 1.60995105628986370670513022320, 1.64352990704299605335983856968, 1.66262567242642535821645263438, 1.66538724268454926844756238908, 1.66952229967608476220811418001, 1.68943499919139246665170233634, 1.75212507987232970629288852219, 1.86359497112076174620157233746, 1.98420257226293536166453664776

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

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