Properties

Degree 20
Conductor $ 2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 10

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 10·7-s − 10·9-s − 10·11-s − 10·13-s − 8·15-s − 3·17-s + 13·19-s − 20·21-s − 14·25-s − 20·27-s − 3·29-s − 31-s − 20·33-s + 40·35-s − 5·37-s − 20·39-s + 4·41-s + 19·43-s + 40·45-s + 47-s + 55·49-s − 6·51-s − 6·53-s + 40·55-s + 26·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 3.77·7-s − 3.33·9-s − 3.01·11-s − 2.77·13-s − 2.06·15-s − 0.727·17-s + 2.98·19-s − 4.36·21-s − 2.79·25-s − 3.84·27-s − 0.557·29-s − 0.179·31-s − 3.48·33-s + 6.76·35-s − 0.821·37-s − 3.20·39-s + 0.624·41-s + 2.89·43-s + 5.96·45-s + 0.145·47-s + 55/7·49-s − 0.840·51-s − 0.824·53-s + 5.39·55-s + 3.44·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  10
Selberg data  =  $(20,\ 2^{30} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 20. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 \( 1 \)
7 \( ( 1 + T )^{10} \)
11 \( ( 1 + T )^{10} \)
13 \( ( 1 + T )^{10} \)
good3 \( 1 - 2 T + 14 T^{2} - 28 T^{3} + 38 p T^{4} - 215 T^{5} + 652 T^{6} - 1120 T^{7} + 938 p T^{8} - 4354 T^{9} + 9530 T^{10} - 4354 p T^{11} + 938 p^{3} T^{12} - 1120 p^{3} T^{13} + 652 p^{4} T^{14} - 215 p^{5} T^{15} + 38 p^{7} T^{16} - 28 p^{7} T^{17} + 14 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
5 \( 1 + 4 T + 6 p T^{2} + 102 T^{3} + 432 T^{4} + 1249 T^{5} + 4028 T^{6} + 10212 T^{7} + 27828 T^{8} + 63842 T^{9} + 153622 T^{10} + 63842 p T^{11} + 27828 p^{2} T^{12} + 10212 p^{3} T^{13} + 4028 p^{4} T^{14} + 1249 p^{5} T^{15} + 432 p^{6} T^{16} + 102 p^{7} T^{17} + 6 p^{9} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 + 3 T + 95 T^{2} + 256 T^{3} + 4627 T^{4} + 11410 T^{5} + 152327 T^{6} + 339784 T^{7} + 3740319 T^{8} + 7485571 T^{9} + 71600122 T^{10} + 7485571 p T^{11} + 3740319 p^{2} T^{12} + 339784 p^{3} T^{13} + 152327 p^{4} T^{14} + 11410 p^{5} T^{15} + 4627 p^{6} T^{16} + 256 p^{7} T^{17} + 95 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 13 T + 189 T^{2} - 1706 T^{3} + 14745 T^{4} - 102986 T^{5} + 667771 T^{6} - 3833770 T^{7} + 20319745 T^{8} - 99300871 T^{9} + 447982818 T^{10} - 99300871 p T^{11} + 20319745 p^{2} T^{12} - 3833770 p^{3} T^{13} + 667771 p^{4} T^{14} - 102986 p^{5} T^{15} + 14745 p^{6} T^{16} - 1706 p^{7} T^{17} + 189 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 105 T^{2} - 56 T^{3} + 6074 T^{4} - 241 p T^{5} + 250282 T^{6} - 280479 T^{7} + 7945845 T^{8} - 9305572 T^{9} + 202470282 T^{10} - 9305572 p T^{11} + 7945845 p^{2} T^{12} - 280479 p^{3} T^{13} + 250282 p^{4} T^{14} - 241 p^{6} T^{15} + 6074 p^{6} T^{16} - 56 p^{7} T^{17} + 105 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 + 3 T + 166 T^{2} + 737 T^{3} + 14060 T^{4} + 73454 T^{5} + 825636 T^{6} + 4263066 T^{7} + 36555311 T^{8} + 168616816 T^{9} + 1223436068 T^{10} + 168616816 p T^{11} + 36555311 p^{2} T^{12} + 4263066 p^{3} T^{13} + 825636 p^{4} T^{14} + 73454 p^{5} T^{15} + 14060 p^{6} T^{16} + 737 p^{7} T^{17} + 166 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + T + 118 T^{2} + 473 T^{3} + 7514 T^{4} + 40895 T^{5} + 418680 T^{6} + 1990441 T^{7} + 18318393 T^{8} + 79682742 T^{9} + 618259844 T^{10} + 79682742 p T^{11} + 18318393 p^{2} T^{12} + 1990441 p^{3} T^{13} + 418680 p^{4} T^{14} + 40895 p^{5} T^{15} + 7514 p^{6} T^{16} + 473 p^{7} T^{17} + 118 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 5 T + 192 T^{2} + 546 T^{3} + 18077 T^{4} + 28360 T^{5} + 1158424 T^{6} + 685440 T^{7} + 56455010 T^{8} + 3760805 T^{9} + 2277323592 T^{10} + 3760805 p T^{11} + 56455010 p^{2} T^{12} + 685440 p^{3} T^{13} + 1158424 p^{4} T^{14} + 28360 p^{5} T^{15} + 18077 p^{6} T^{16} + 546 p^{7} T^{17} + 192 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 - 4 T + 278 T^{2} - 797 T^{3} + 35548 T^{4} - 67687 T^{5} + 2842396 T^{6} - 3261309 T^{7} + 164091039 T^{8} - 2803939 p T^{9} + 7459923348 T^{10} - 2803939 p^{2} T^{11} + 164091039 p^{2} T^{12} - 3261309 p^{3} T^{13} + 2842396 p^{4} T^{14} - 67687 p^{5} T^{15} + 35548 p^{6} T^{16} - 797 p^{7} T^{17} + 278 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 19 T + 438 T^{2} - 5579 T^{3} + 78112 T^{4} - 777998 T^{5} + 8241420 T^{6} - 67747847 T^{7} + 585058732 T^{8} - 4067915183 T^{9} + 29578624826 T^{10} - 4067915183 p T^{11} + 585058732 p^{2} T^{12} - 67747847 p^{3} T^{13} + 8241420 p^{4} T^{14} - 777998 p^{5} T^{15} + 78112 p^{6} T^{16} - 5579 p^{7} T^{17} + 438 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - T + 262 T^{2} + 153 T^{3} + 30894 T^{4} + 83172 T^{5} + 2165372 T^{6} + 12067076 T^{7} + 107046149 T^{8} + 936776498 T^{9} + 4811878804 T^{10} + 936776498 p T^{11} + 107046149 p^{2} T^{12} + 12067076 p^{3} T^{13} + 2165372 p^{4} T^{14} + 83172 p^{5} T^{15} + 30894 p^{6} T^{16} + 153 p^{7} T^{17} + 262 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 + 6 T + 262 T^{2} + 1050 T^{3} + 25644 T^{4} + 61244 T^{5} + 1085656 T^{6} + 1138714 T^{7} + 6402542 T^{8} - 13499098 T^{9} - 1084113186 T^{10} - 13499098 p T^{11} + 6402542 p^{2} T^{12} + 1138714 p^{3} T^{13} + 1085656 p^{4} T^{14} + 61244 p^{5} T^{15} + 25644 p^{6} T^{16} + 1050 p^{7} T^{17} + 262 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 4 T + 376 T^{2} - 542 T^{3} + 62849 T^{4} + 67055 T^{5} + 6391122 T^{6} + 385777 p T^{7} + 468791946 T^{8} + 2519238602 T^{9} + 28958736180 T^{10} + 2519238602 p T^{11} + 468791946 p^{2} T^{12} + 385777 p^{4} T^{13} + 6391122 p^{4} T^{14} + 67055 p^{5} T^{15} + 62849 p^{6} T^{16} - 542 p^{7} T^{17} + 376 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 18 T + 579 T^{2} + 7993 T^{3} + 146405 T^{4} + 1650448 T^{5} + 22082673 T^{6} + 209469501 T^{7} + 2240011157 T^{8} + 296416790 p T^{9} + 161060962142 T^{10} + 296416790 p^{2} T^{11} + 2240011157 p^{2} T^{12} + 209469501 p^{3} T^{13} + 22082673 p^{4} T^{14} + 1650448 p^{5} T^{15} + 146405 p^{6} T^{16} + 7993 p^{7} T^{17} + 579 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - T + 323 T^{2} - 496 T^{3} + 56629 T^{4} - 104683 T^{5} + 6895709 T^{6} - 14257376 T^{7} + 643359771 T^{8} - 1339953987 T^{9} + 47989270290 T^{10} - 1339953987 p T^{11} + 643359771 p^{2} T^{12} - 14257376 p^{3} T^{13} + 6895709 p^{4} T^{14} - 104683 p^{5} T^{15} + 56629 p^{6} T^{16} - 496 p^{7} T^{17} + 323 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 21 T + 595 T^{2} + 9050 T^{3} + 153296 T^{4} + 1846290 T^{5} + 23670438 T^{6} + 238641380 T^{7} + 2537102703 T^{8} + 22180291527 T^{9} + 204716901198 T^{10} + 22180291527 p T^{11} + 2537102703 p^{2} T^{12} + 238641380 p^{3} T^{13} + 23670438 p^{4} T^{14} + 1846290 p^{5} T^{15} + 153296 p^{6} T^{16} + 9050 p^{7} T^{17} + 595 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 + 10 T + 430 T^{2} + 3123 T^{3} + 84218 T^{4} + 459825 T^{5} + 10645064 T^{6} + 45251323 T^{7} + 1024123009 T^{8} + 3639285855 T^{9} + 81388231564 T^{10} + 3639285855 p T^{11} + 1024123009 p^{2} T^{12} + 45251323 p^{3} T^{13} + 10645064 p^{4} T^{14} + 459825 p^{5} T^{15} + 84218 p^{6} T^{16} + 3123 p^{7} T^{17} + 430 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 3 T + 254 T^{2} + 531 T^{3} + 31102 T^{4} + 158392 T^{5} + 3948602 T^{6} + 17160001 T^{7} + 400678518 T^{8} + 2118028899 T^{9} + 31049075910 T^{10} + 2118028899 p T^{11} + 400678518 p^{2} T^{12} + 17160001 p^{3} T^{13} + 3948602 p^{4} T^{14} + 158392 p^{5} T^{15} + 31102 p^{6} T^{16} + 531 p^{7} T^{17} + 254 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 6 T + 539 T^{2} - 4201 T^{3} + 143119 T^{4} - 1270904 T^{5} + 24914087 T^{6} - 227115441 T^{7} + 3148075691 T^{8} - 27015619936 T^{9} + 299677355034 T^{10} - 27015619936 p T^{11} + 3148075691 p^{2} T^{12} - 227115441 p^{3} T^{13} + 24914087 p^{4} T^{14} - 1270904 p^{5} T^{15} + 143119 p^{6} T^{16} - 4201 p^{7} T^{17} + 539 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 + 22 T + 656 T^{2} + 12304 T^{3} + 222916 T^{4} + 3358293 T^{5} + 47532416 T^{6} + 590926676 T^{7} + 6925796868 T^{8} + 72929260954 T^{9} + 722999129492 T^{10} + 72929260954 p T^{11} + 6925796868 p^{2} T^{12} + 590926676 p^{3} T^{13} + 47532416 p^{4} T^{14} + 3358293 p^{5} T^{15} + 222916 p^{6} T^{16} + 12304 p^{7} T^{17} + 656 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 9 T + 736 T^{2} + 58 p T^{3} + 257179 T^{4} + 1707422 T^{5} + 56621714 T^{6} + 329089196 T^{7} + 8722674276 T^{8} + 44244616163 T^{9} + 983033276348 T^{10} + 44244616163 p T^{11} + 8722674276 p^{2} T^{12} + 329089196 p^{3} T^{13} + 56621714 p^{4} T^{14} + 1707422 p^{5} T^{15} + 257179 p^{6} T^{16} + 58 p^{8} T^{17} + 736 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.03246872263447133645241536220, −2.99372050873294885968319230066, −2.94452849904266848381972655043, −2.91481707441893768589152126901, −2.83621684703879289783438041372, −2.57731296082003771467397346684, −2.51517401603686196991246059394, −2.50986716267567213031602799535, −2.45894774798311388434405949580, −2.32571504312232413714910215847, −2.32497437985182095910618949073, −2.27400241457980682878498697905, −2.23653456983946074040187151589, −2.19527184306508229630788350721, −2.15449568593450195918357390586, −1.77988447269820424813305028915, −1.49167266948210717582785077407, −1.44623589490272966392194170247, −1.43764328632949055633781768952, −1.35214261081046446943092972678, −1.12970028524986526897660205189, −0.972757017434179077151144494539, −0.968200407501423047824460775005, −0.914648945021387609450356470762, −0.77607278394989033482329017714, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77607278394989033482329017714, 0.914648945021387609450356470762, 0.968200407501423047824460775005, 0.972757017434179077151144494539, 1.12970028524986526897660205189, 1.35214261081046446943092972678, 1.43764328632949055633781768952, 1.44623589490272966392194170247, 1.49167266948210717582785077407, 1.77988447269820424813305028915, 2.15449568593450195918357390586, 2.19527184306508229630788350721, 2.23653456983946074040187151589, 2.27400241457980682878498697905, 2.32497437985182095910618949073, 2.32571504312232413714910215847, 2.45894774798311388434405949580, 2.50986716267567213031602799535, 2.51517401603686196991246059394, 2.57731296082003771467397346684, 2.83621684703879289783438041372, 2.91481707441893768589152126901, 2.94452849904266848381972655043, 2.99372050873294885968319230066, 3.03246872263447133645241536220

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

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