Properties

Label 8-6762e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.091\times 10^{15}$
Sign $1$
Analytic cond. $8.49980\times 10^{6}$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 10·4-s − 4·5-s + 16·6-s + 20·8-s + 10·9-s − 16·10-s + 6·11-s + 40·12-s + 6·13-s − 16·15-s + 35·16-s − 10·17-s + 40·18-s − 4·19-s − 40·20-s + 24·22-s + 4·23-s + 80·24-s + 6·25-s + 24·26-s + 20·27-s + 6·29-s − 64·30-s + 2·31-s + 56·32-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 7.07·8-s + 10/3·9-s − 5.05·10-s + 1.80·11-s + 11.5·12-s + 1.66·13-s − 4.13·15-s + 35/4·16-s − 2.42·17-s + 9.42·18-s − 0.917·19-s − 8.94·20-s + 5.11·22-s + 0.834·23-s + 16.3·24-s + 6/5·25-s + 4.70·26-s + 3.84·27-s + 1.11·29-s − 11.6·30-s + 0.359·31-s + 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(8.49980\times 10^{6}\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6762} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(197.3523797\)
\(L(\frac12)\) \(\approx\) \(197.3523797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$D_{4}$ \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 25 T^{2} - 42 T^{3} + 120 T^{4} - 42 p T^{5} + 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 51 T^{2} - 198 T^{3} + 986 T^{4} - 198 p T^{5} + 51 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 83 T^{2} + 26 p T^{3} + 2098 T^{4} + 26 p^{2} T^{5} + 83 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 46 T^{2} + 196 T^{3} + 1090 T^{4} + 196 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 115 T^{2} - 498 T^{3} + 5004 T^{4} - 498 p T^{5} + 115 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 57 T^{2} - 406 T^{3} + 1424 T^{4} - 406 p T^{5} + 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 90 T^{2} + 72 T^{3} + 4058 T^{4} + 72 p T^{5} + 90 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 134 T^{2} - 460 T^{3} + 7690 T^{4} - 460 p T^{5} + 134 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 192 T^{2} - 28 p T^{3} + 7118 T^{4} - 28 p^{2} T^{5} + 192 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 113 T^{2} - 910 T^{3} + 7552 T^{4} - 910 p T^{5} + 113 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 154 T^{2} + 48 T^{3} + 11259 T^{4} + 48 p T^{5} + 154 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 187 T^{2} - 846 T^{3} + 15216 T^{4} - 846 p T^{5} + 187 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 12 T^{2} + 384 T^{3} + 2870 T^{4} + 384 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 169 T^{2} - 630 T^{3} + 2472 T^{4} - 630 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 42 T + 931 T^{2} - 13278 T^{3} + 132774 T^{4} - 13278 p T^{5} + 931 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 237 T^{2} - 1410 T^{3} + 23912 T^{4} - 1410 p T^{5} + 237 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 315 T^{2} + 1398 T^{3} + 37304 T^{4} + 1398 p T^{5} + 315 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 241 T^{2} + 2270 T^{3} + 27480 T^{4} + 2270 p T^{5} + 241 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 250 T^{2} + 504 T^{3} + 28026 T^{4} + 504 p T^{5} + 250 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 363 T^{2} + 2702 T^{3} + 51668 T^{4} + 2702 p T^{5} + 363 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.64333648399509723370690061008, −5.20463603078767512602805686309, −4.95246240546459578889603385890, −4.93033541121746925577249974103, −4.76214164818281453252373969762, −4.34902079629440952627410384293, −4.26676312830454856589300227645, −4.23457163350382627171843557473, −4.15816002535406300806085538064, −3.74412904500221142071997334264, −3.69545022082359932926710042491, −3.67088411558871588578206117885, −3.59988632847214976172859410917, −3.18151716582751451571947937862, −2.88306889019999387035211139889, −2.85880406239217141672989474859, −2.46588704901395158947411489153, −2.27409989565074935112491614219, −2.23375557849041268040594889810, −1.90905534449666536080470170231, −1.90830249315047160082654340972, −1.17850285262045586183958881755, −1.05718014400688246872445867769, −0.850053492920273040032462580801, −0.61053783552254470587351278637, 0.61053783552254470587351278637, 0.850053492920273040032462580801, 1.05718014400688246872445867769, 1.17850285262045586183958881755, 1.90830249315047160082654340972, 1.90905534449666536080470170231, 2.23375557849041268040594889810, 2.27409989565074935112491614219, 2.46588704901395158947411489153, 2.85880406239217141672989474859, 2.88306889019999387035211139889, 3.18151716582751451571947937862, 3.59988632847214976172859410917, 3.67088411558871588578206117885, 3.69545022082359932926710042491, 3.74412904500221142071997334264, 4.15816002535406300806085538064, 4.23457163350382627171843557473, 4.26676312830454856589300227645, 4.34902079629440952627410384293, 4.76214164818281453252373969762, 4.93033541121746925577249974103, 4.95246240546459578889603385890, 5.20463603078767512602805686309, 5.64333648399509723370690061008

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