Properties

Degree 16
Conductor $ 2^{16} \cdot 19^{8} \cdot 79^{8} $
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  + 4·5-s + 7-s − 24·9-s − 7·11-s − 12·13-s − 7·17-s + 8·19-s + 10·23-s − 9·25-s + 5·29-s + 3·31-s + 4·35-s − 15·37-s − 5·41-s + 10·43-s − 96·45-s + 18·47-s − 12·49-s + 9·53-s − 28·55-s + 8·59-s + 13·61-s − 24·63-s − 48·65-s + 21·67-s + 44·71-s − 20·73-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s − 8·9-s − 2.11·11-s − 3.32·13-s − 1.69·17-s + 1.83·19-s + 2.08·23-s − 9/5·25-s + 0.928·29-s + 0.538·31-s + 0.676·35-s − 2.46·37-s − 0.780·41-s + 1.52·43-s − 14.3·45-s + 2.62·47-s − 1.71·49-s + 1.23·53-s − 3.77·55-s + 1.04·59-s + 1.66·61-s − 3.02·63-s − 5.95·65-s + 2.56·67-s + 5.22·71-s − 2.34·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8} \cdot 79^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8} \cdot 79^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{16} \cdot 19^{8} \cdot 79^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{16} \cdot 19^{8} \cdot 79^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $7.763120546$
$L(\frac12)$  $\approx$  $7.763120546$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( 1 \)
19 \( ( 1 - T )^{8} \)
79 \( ( 1 - T )^{8} \)
good3 \( ( 1 + p T^{2} )^{8} \)
5 \( 1 - 4 T + p^{2} T^{2} - 84 T^{3} + 337 T^{4} - 948 T^{5} + 2874 T^{6} - 1376 p T^{7} + 17138 T^{8} - 1376 p^{2} T^{9} + 2874 p^{2} T^{10} - 948 p^{3} T^{11} + 337 p^{4} T^{12} - 84 p^{5} T^{13} + p^{8} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - T + 13 T^{2} - 16 T^{3} + 96 T^{4} - 40 T^{5} + 411 T^{6} + 97 p T^{7} + 1814 T^{8} + 97 p^{2} T^{9} + 411 p^{2} T^{10} - 40 p^{3} T^{11} + 96 p^{4} T^{12} - 16 p^{5} T^{13} + 13 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 7 T + 45 T^{2} + 240 T^{3} + 1092 T^{4} + 4810 T^{5} + 19147 T^{6} + 69981 T^{7} + 249270 T^{8} + 69981 p T^{9} + 19147 p^{2} T^{10} + 4810 p^{3} T^{11} + 1092 p^{4} T^{12} + 240 p^{5} T^{13} + 45 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 12 T + 8 p T^{2} + 670 T^{3} + 3932 T^{4} + 20094 T^{5} + 93464 T^{6} + 382864 T^{7} + 1450726 T^{8} + 382864 p T^{9} + 93464 p^{2} T^{10} + 20094 p^{3} T^{11} + 3932 p^{4} T^{12} + 670 p^{5} T^{13} + 8 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 7 T + 103 T^{2} + 548 T^{3} + 4706 T^{4} + 20574 T^{5} + 133017 T^{6} + 495463 T^{7} + 2645882 T^{8} + 495463 p T^{9} + 133017 p^{2} T^{10} + 20574 p^{3} T^{11} + 4706 p^{4} T^{12} + 548 p^{5} T^{13} + 103 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 10 T + 148 T^{2} - 50 p T^{3} + 10460 T^{4} - 65118 T^{5} + 443372 T^{6} - 2270570 T^{7} + 12421126 T^{8} - 2270570 p T^{9} + 443372 p^{2} T^{10} - 65118 p^{3} T^{11} + 10460 p^{4} T^{12} - 50 p^{6} T^{13} + 148 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 5 T + 123 T^{2} - 632 T^{3} + 8076 T^{4} - 40332 T^{5} + 367881 T^{6} - 1687507 T^{7} + 12353870 T^{8} - 1687507 p T^{9} + 367881 p^{2} T^{10} - 40332 p^{3} T^{11} + 8076 p^{4} T^{12} - 632 p^{5} T^{13} + 123 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 3 T + 85 T^{2} - 104 T^{3} + 4332 T^{4} - 2254 T^{5} + 173379 T^{6} + 53815 T^{7} + 5357270 T^{8} + 53815 p T^{9} + 173379 p^{2} T^{10} - 2254 p^{3} T^{11} + 4332 p^{4} T^{12} - 104 p^{5} T^{13} + 85 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 15 T + 237 T^{2} + 2302 T^{3} + 23122 T^{4} + 175446 T^{5} + 1383461 T^{6} + 8836885 T^{7} + 59177890 T^{8} + 8836885 p T^{9} + 1383461 p^{2} T^{10} + 175446 p^{3} T^{11} + 23122 p^{4} T^{12} + 2302 p^{5} T^{13} + 237 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 5 T + 191 T^{2} + 866 T^{3} + 17988 T^{4} + 72014 T^{5} + 1116991 T^{6} + 3913103 T^{7} + 51879078 T^{8} + 3913103 p T^{9} + 1116991 p^{2} T^{10} + 72014 p^{3} T^{11} + 17988 p^{4} T^{12} + 866 p^{5} T^{13} + 191 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 10 T + 295 T^{2} - 2540 T^{3} + 39711 T^{4} - 293262 T^{5} + 3191662 T^{6} - 19896490 T^{7} + 167715516 T^{8} - 19896490 p T^{9} + 3191662 p^{2} T^{10} - 293262 p^{3} T^{11} + 39711 p^{4} T^{12} - 2540 p^{5} T^{13} + 295 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 18 T + 263 T^{2} - 2560 T^{3} + 25259 T^{4} - 188914 T^{5} + 1472786 T^{6} - 9491518 T^{7} + 71064236 T^{8} - 9491518 p T^{9} + 1472786 p^{2} T^{10} - 188914 p^{3} T^{11} + 25259 p^{4} T^{12} - 2560 p^{5} T^{13} + 263 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 9 T + 181 T^{2} - 464 T^{3} + 10156 T^{4} + 7868 T^{5} + 802565 T^{6} - 1836179 T^{7} + 63746742 T^{8} - 1836179 p T^{9} + 802565 p^{2} T^{10} + 7868 p^{3} T^{11} + 10156 p^{4} T^{12} - 464 p^{5} T^{13} + 181 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 8 T + 252 T^{2} - 1240 T^{3} + 26964 T^{4} - 107256 T^{5} + 2224100 T^{6} - 9475496 T^{7} + 154140598 T^{8} - 9475496 p T^{9} + 2224100 p^{2} T^{10} - 107256 p^{3} T^{11} + 26964 p^{4} T^{12} - 1240 p^{5} T^{13} + 252 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 13 T + 413 T^{2} - 4122 T^{3} + 76238 T^{4} - 621696 T^{5} + 8486267 T^{6} - 57502613 T^{7} + 627276258 T^{8} - 57502613 p T^{9} + 8486267 p^{2} T^{10} - 621696 p^{3} T^{11} + 76238 p^{4} T^{12} - 4122 p^{5} T^{13} + 413 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 21 T + 647 T^{2} - 9526 T^{3} + 166524 T^{4} - 1867294 T^{5} + 23315863 T^{6} - 205726833 T^{7} + 1978152066 T^{8} - 205726833 p T^{9} + 23315863 p^{2} T^{10} - 1867294 p^{3} T^{11} + 166524 p^{4} T^{12} - 9526 p^{5} T^{13} + 647 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 44 T + 1282 T^{2} - 26938 T^{3} + 456972 T^{4} - 6426094 T^{5} + 77190686 T^{6} - 11267920 p T^{7} + 7221053478 T^{8} - 11267920 p^{2} T^{9} + 77190686 p^{2} T^{10} - 6426094 p^{3} T^{11} + 456972 p^{4} T^{12} - 26938 p^{5} T^{13} + 1282 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 20 T + 577 T^{2} + 9036 T^{3} + 146493 T^{4} + 1823236 T^{5} + 21290890 T^{6} + 213010104 T^{7} + 1939271878 T^{8} + 213010104 p T^{9} + 21290890 p^{2} T^{10} + 1823236 p^{3} T^{11} + 146493 p^{4} T^{12} + 9036 p^{5} T^{13} + 577 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 4 T + 576 T^{2} + 2392 T^{3} + 149900 T^{4} + 606992 T^{5} + 23200192 T^{6} + 84755532 T^{7} + 2349690630 T^{8} + 84755532 p T^{9} + 23200192 p^{2} T^{10} + 606992 p^{3} T^{11} + 149900 p^{4} T^{12} + 2392 p^{5} T^{13} + 576 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 10 T + 432 T^{2} + 4376 T^{3} + 97892 T^{4} + 937012 T^{5} + 14548976 T^{6} + 124904378 T^{7} + 1524844726 T^{8} + 124904378 p T^{9} + 14548976 p^{2} T^{10} + 937012 p^{3} T^{11} + 97892 p^{4} T^{12} + 4376 p^{5} T^{13} + 432 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 22 T + 760 T^{2} + 11450 T^{3} + 229228 T^{4} + 2655822 T^{5} + 39640072 T^{6} + 375560034 T^{7} + 4585904678 T^{8} + 375560034 p T^{9} + 39640072 p^{2} T^{10} + 2655822 p^{3} T^{11} + 229228 p^{4} T^{12} + 11450 p^{5} T^{13} + 760 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.29275946251221900391537076247, −3.06703124627373126351930895887, −2.89363710043659577501309702926, −2.81361008596563582629856639807, −2.72522191933860605380199858900, −2.67392106407307981181971101220, −2.67131906968221753793305778868, −2.63707884384862693490496825964, −2.53632258382607028270638653778, −2.28029592204728095805483094219, −2.16330244336412494014137388729, −2.01934882041315541065709682890, −2.00776341032932074829068642931, −1.98813996298801293998505549488, −1.92189871611579277591957116742, −1.82056650467146323526691565254, −1.71708694913121798316435329284, −1.07198056032298388612842116247, −0.878574187835565482688778706510, −0.78437141148040842709414059503, −0.49298851118096174713315771942, −0.47813055023363177415163328192, −0.46562500952041516561778198642, −0.41258655127817816240804506498, −0.40470571370775683434275271680, 0.40470571370775683434275271680, 0.41258655127817816240804506498, 0.46562500952041516561778198642, 0.47813055023363177415163328192, 0.49298851118096174713315771942, 0.78437141148040842709414059503, 0.878574187835565482688778706510, 1.07198056032298388612842116247, 1.71708694913121798316435329284, 1.82056650467146323526691565254, 1.92189871611579277591957116742, 1.98813996298801293998505549488, 2.00776341032932074829068642931, 2.01934882041315541065709682890, 2.16330244336412494014137388729, 2.28029592204728095805483094219, 2.53632258382607028270638653778, 2.63707884384862693490496825964, 2.67131906968221753793305778868, 2.67392106407307981181971101220, 2.72522191933860605380199858900, 2.81361008596563582629856639807, 2.89363710043659577501309702926, 3.06703124627373126351930895887, 3.29275946251221900391537076247

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

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