Properties

Degree 20
Conductor $ 3^{20} \cdot 7^{10} \cdot 11^{20} $
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  − 4-s − 5·5-s + 10·7-s + 8-s + 6·13-s + 16-s − 8·17-s + 5·20-s + 3·25-s − 10·28-s + 14·29-s + 26·31-s + 6·32-s − 50·35-s + 24·37-s − 5·40-s − 19·41-s − 6·43-s − 15·47-s + 55·49-s − 6·52-s + 53-s + 10·56-s − 23·59-s − 4·64-s − 30·65-s + 38·67-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.23·5-s + 3.77·7-s + 0.353·8-s + 1.66·13-s + 1/4·16-s − 1.94·17-s + 1.11·20-s + 3/5·25-s − 1.88·28-s + 2.59·29-s + 4.66·31-s + 1.06·32-s − 8.45·35-s + 3.94·37-s − 0.790·40-s − 2.96·41-s − 0.914·43-s − 2.18·47-s + 55/7·49-s − 0.832·52-s + 0.137·53-s + 1.33·56-s − 2.99·59-s − 1/2·64-s − 3.72·65-s + 4.64·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(3^{20} \cdot 7^{10} \cdot 11^{20}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 3^{20} \cdot 7^{10} \cdot 11^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $\approx$  $41.43178652$
$L(\frac12)$  $\approx$  $41.43178652$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 20. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - T )^{10} \)
11 \( 1 \)
good2 \( 1 + T^{2} - T^{3} - p^{3} T^{5} + p^{2} T^{6} - 7 T^{7} + 13 T^{8} - p^{3} T^{9} + 41 T^{10} - p^{4} T^{11} + 13 p^{2} T^{12} - 7 p^{3} T^{13} + p^{6} T^{14} - p^{8} T^{15} - p^{7} T^{17} + p^{8} T^{18} + p^{10} T^{20} \)
5 \( 1 + p T + 22 T^{2} + 46 T^{3} + 113 T^{4} + 108 T^{5} + 391 T^{6} + 538 T^{7} + 4002 T^{8} + 9087 T^{9} + 31806 T^{10} + 9087 p T^{11} + 4002 p^{2} T^{12} + 538 p^{3} T^{13} + 391 p^{4} T^{14} + 108 p^{5} T^{15} + 113 p^{6} T^{16} + 46 p^{7} T^{17} + 22 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \)
13 \( 1 - 6 T + 5 p T^{2} - 368 T^{3} + 2385 T^{4} - 10918 T^{5} + 56756 T^{6} - 221994 T^{7} + 984630 T^{8} - 3474482 T^{9} + 13929014 T^{10} - 3474482 p T^{11} + 984630 p^{2} T^{12} - 221994 p^{3} T^{13} + 56756 p^{4} T^{14} - 10918 p^{5} T^{15} + 2385 p^{6} T^{16} - 368 p^{7} T^{17} + 5 p^{9} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 + 8 T + 8 p T^{2} + 954 T^{3} + 9093 T^{4} + 54096 T^{5} + 378365 T^{6} + 1910556 T^{7} + 10713786 T^{8} + 46048194 T^{9} + 215035238 T^{10} + 46048194 p T^{11} + 10713786 p^{2} T^{12} + 1910556 p^{3} T^{13} + 378365 p^{4} T^{14} + 54096 p^{5} T^{15} + 9093 p^{6} T^{16} + 954 p^{7} T^{17} + 8 p^{9} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 69 T^{2} + 136 T^{3} + 2758 T^{4} + 8362 T^{5} + 89118 T^{6} + 289638 T^{7} + 2274617 T^{8} + 397140 p T^{9} + 46843450 T^{10} + 397140 p^{2} T^{11} + 2274617 p^{2} T^{12} + 289638 p^{3} T^{13} + 89118 p^{4} T^{14} + 8362 p^{5} T^{15} + 2758 p^{6} T^{16} + 136 p^{7} T^{17} + 69 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 + 124 T^{2} - 38 T^{3} + 8330 T^{4} - 158 p T^{5} + 381506 T^{6} - 180342 T^{7} + 12965000 T^{8} - 5936040 T^{9} + 338330610 T^{10} - 5936040 p T^{11} + 12965000 p^{2} T^{12} - 180342 p^{3} T^{13} + 381506 p^{4} T^{14} - 158 p^{6} T^{15} + 8330 p^{6} T^{16} - 38 p^{7} T^{17} + 124 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 - 14 T + 200 T^{2} - 1928 T^{3} + 18575 T^{4} - 144068 T^{5} + 1110481 T^{6} - 7337936 T^{7} + 48191364 T^{8} - 277661302 T^{9} + 1587859174 T^{10} - 277661302 p T^{11} + 48191364 p^{2} T^{12} - 7337936 p^{3} T^{13} + 1110481 p^{4} T^{14} - 144068 p^{5} T^{15} + 18575 p^{6} T^{16} - 1928 p^{7} T^{17} + 200 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 26 T + 482 T^{2} - 6400 T^{3} + 71453 T^{4} - 670846 T^{5} + 5624329 T^{6} - 41976980 T^{7} + 287918626 T^{8} - 1800633596 T^{9} + 10465823578 T^{10} - 1800633596 p T^{11} + 287918626 p^{2} T^{12} - 41976980 p^{3} T^{13} + 5624329 p^{4} T^{14} - 670846 p^{5} T^{15} + 71453 p^{6} T^{16} - 6400 p^{7} T^{17} + 482 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 24 T + 440 T^{2} - 5840 T^{3} + 65456 T^{4} - 634664 T^{5} + 5522032 T^{6} - 43933344 T^{7} + 323265704 T^{8} - 2198445368 T^{9} + 13921174566 T^{10} - 2198445368 p T^{11} + 323265704 p^{2} T^{12} - 43933344 p^{3} T^{13} + 5522032 p^{4} T^{14} - 634664 p^{5} T^{15} + 65456 p^{6} T^{16} - 5840 p^{7} T^{17} + 440 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 19 T + 484 T^{2} + 6388 T^{3} + 94475 T^{4} + 963532 T^{5} + 10457893 T^{6} + 86623352 T^{7} + 748989936 T^{8} + 5154844493 T^{9} + 36776727094 T^{10} + 5154844493 p T^{11} + 748989936 p^{2} T^{12} + 86623352 p^{3} T^{13} + 10457893 p^{4} T^{14} + 963532 p^{5} T^{15} + 94475 p^{6} T^{16} + 6388 p^{7} T^{17} + 484 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 6 T + 312 T^{2} + 1828 T^{3} + 47319 T^{4} + 259736 T^{5} + 4576717 T^{6} + 22838020 T^{7} + 311402464 T^{8} + 1377461130 T^{9} + 15540959670 T^{10} + 1377461130 p T^{11} + 311402464 p^{2} T^{12} + 22838020 p^{3} T^{13} + 4576717 p^{4} T^{14} + 259736 p^{5} T^{15} + 47319 p^{6} T^{16} + 1828 p^{7} T^{17} + 312 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 15 T + 333 T^{2} + 4161 T^{3} + 56109 T^{4} + 576028 T^{5} + 127236 p T^{6} + 51937604 T^{7} + 445599026 T^{8} + 3323250890 T^{9} + 24323199470 T^{10} + 3323250890 p T^{11} + 445599026 p^{2} T^{12} + 51937604 p^{3} T^{13} + 127236 p^{5} T^{14} + 576028 p^{5} T^{15} + 56109 p^{6} T^{16} + 4161 p^{7} T^{17} + 333 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - T + 254 T^{2} + 378 T^{3} + 28817 T^{4} + 120372 T^{5} + 2050339 T^{6} + 14991722 T^{7} + 109654718 T^{8} + 1160429241 T^{9} + 5512758766 T^{10} + 1160429241 p T^{11} + 109654718 p^{2} T^{12} + 14991722 p^{3} T^{13} + 2050339 p^{4} T^{14} + 120372 p^{5} T^{15} + 28817 p^{6} T^{16} + 378 p^{7} T^{17} + 254 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 + 23 T + 495 T^{2} + 6037 T^{3} + 71081 T^{4} + 556772 T^{5} + 4700924 T^{6} + 26168124 T^{7} + 209482574 T^{8} + 1016636482 T^{9} + 10462754634 T^{10} + 1016636482 p T^{11} + 209482574 p^{2} T^{12} + 26168124 p^{3} T^{13} + 4700924 p^{4} T^{14} + 556772 p^{5} T^{15} + 71081 p^{6} T^{16} + 6037 p^{7} T^{17} + 495 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 215 T^{2} - 490 T^{3} + 23725 T^{4} - 88654 T^{5} + 1906100 T^{6} - 8267970 T^{7} + 125871770 T^{8} - 598496950 T^{9} + 7588984906 T^{10} - 598496950 p T^{11} + 125871770 p^{2} T^{12} - 8267970 p^{3} T^{13} + 1906100 p^{4} T^{14} - 88654 p^{5} T^{15} + 23725 p^{6} T^{16} - 490 p^{7} T^{17} + 215 p^{8} T^{18} + p^{10} T^{20} \)
67 \( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 33446868814 p T^{11} + 3720320353 p^{2} T^{12} - 384823180 p^{3} T^{13} + 37034650 p^{4} T^{14} - 3231648 p^{5} T^{15} + 253022 p^{6} T^{16} - 16920 p^{7} T^{17} + 939 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 26 T + 632 T^{2} + 9130 T^{3} + 126663 T^{4} + 1295526 T^{5} + 13933979 T^{6} + 123894450 T^{7} + 1257395276 T^{8} + 10721089596 T^{9} + 100241082738 T^{10} + 10721089596 p T^{11} + 1257395276 p^{2} T^{12} + 123894450 p^{3} T^{13} + 13933979 p^{4} T^{14} + 1295526 p^{5} T^{15} + 126663 p^{6} T^{16} + 9130 p^{7} T^{17} + 632 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 + T + 393 T^{2} + 817 T^{3} + 74513 T^{4} + 250396 T^{5} + 9185572 T^{6} + 43089004 T^{7} + 854063166 T^{8} + 4726643438 T^{9} + 66584553318 T^{10} + 4726643438 p T^{11} + 854063166 p^{2} T^{12} + 43089004 p^{3} T^{13} + 9185572 p^{4} T^{14} + 250396 p^{5} T^{15} + 74513 p^{6} T^{16} + 817 p^{7} T^{17} + 393 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 5 T + 552 T^{2} - 2604 T^{3} + 151027 T^{4} - 658718 T^{5} + 26569829 T^{6} - 105054020 T^{7} + 3304178864 T^{8} - 11590016269 T^{9} + 302495065710 T^{10} - 11590016269 p T^{11} + 3304178864 p^{2} T^{12} - 105054020 p^{3} T^{13} + 26569829 p^{4} T^{14} - 658718 p^{5} T^{15} + 151027 p^{6} T^{16} - 2604 p^{7} T^{17} + 552 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 + 6 T + 490 T^{2} + 3578 T^{3} + 118805 T^{4} + 976848 T^{5} + 19084536 T^{6} + 164833504 T^{7} + 2271305170 T^{8} + 19148739052 T^{9} + 211320377724 T^{10} + 19148739052 p T^{11} + 2271305170 p^{2} T^{12} + 164833504 p^{3} T^{13} + 19084536 p^{4} T^{14} + 976848 p^{5} T^{15} + 118805 p^{6} T^{16} + 3578 p^{7} T^{17} + 490 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 9 T + 640 T^{2} - 4050 T^{3} + 178355 T^{4} - 698876 T^{5} + 29313265 T^{6} - 51732294 T^{7} + 3369419368 T^{8} - 1036172651 T^{9} + 317703552166 T^{10} - 1036172651 p T^{11} + 3369419368 p^{2} T^{12} - 51732294 p^{3} T^{13} + 29313265 p^{4} T^{14} - 698876 p^{5} T^{15} + 178355 p^{6} T^{16} - 4050 p^{7} T^{17} + 640 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 24 T + 1069 T^{2} - 19604 T^{3} + 490777 T^{4} - 7233428 T^{5} + 130302052 T^{6} - 1582958620 T^{7} + 22393814246 T^{8} - 226465922980 T^{9} + 2616113341790 T^{10} - 226465922980 p T^{11} + 22393814246 p^{2} T^{12} - 1582958620 p^{3} T^{13} + 130302052 p^{4} T^{14} - 7233428 p^{5} T^{15} + 490777 p^{6} T^{16} - 19604 p^{7} T^{17} + 1069 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
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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

−2.73004268311064683472228196863, −2.40156481697510883925838139051, −2.38373536755983231126548800316, −2.35138274430449711104991069799, −2.33952851171055968153378125752, −2.26600938885505864241736041076, −2.20974488078276763235920640659, −2.17363011524403984393700531652, −1.77953597960529035707312861298, −1.68407714579041391185556476754, −1.57990259056220964155235500442, −1.56317046483898476247329158466, −1.52322090420873578241691054630, −1.50661950918317011526586963203, −1.46795974368203034391901206849, −1.31511371623351703495988499243, −0.985139180003050969382983945775, −0.976339835528902820581426339494, −0.904534672416759829732917227783, −0.72702679855630541874826691832, −0.68092491107130066562941294686, −0.58767375107835252409120330919, −0.44004941168209716956979192531, −0.34266297581859492527457302155, −0.20166689580682992814219565898, 0.20166689580682992814219565898, 0.34266297581859492527457302155, 0.44004941168209716956979192531, 0.58767375107835252409120330919, 0.68092491107130066562941294686, 0.72702679855630541874826691832, 0.904534672416759829732917227783, 0.976339835528902820581426339494, 0.985139180003050969382983945775, 1.31511371623351703495988499243, 1.46795974368203034391901206849, 1.50661950918317011526586963203, 1.52322090420873578241691054630, 1.56317046483898476247329158466, 1.57990259056220964155235500442, 1.68407714579041391185556476754, 1.77953597960529035707312861298, 2.17363011524403984393700531652, 2.20974488078276763235920640659, 2.26600938885505864241736041076, 2.33952851171055968153378125752, 2.35138274430449711104991069799, 2.38373536755983231126548800316, 2.40156481697510883925838139051, 2.73004268311064683472228196863

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

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