Properties

Label 8-1960e4-1.1-c1e4-0-7
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s − 9-s + 2·11-s + 10·13-s + 8·15-s + 6·17-s − 4·23-s + 10·25-s − 2·27-s − 2·29-s − 12·31-s + 4·33-s + 20·39-s + 12·41-s − 8·43-s − 4·45-s − 2·47-s + 12·51-s − 4·53-s + 8·55-s + 8·59-s + 20·61-s + 40·65-s − 8·67-s − 8·69-s + 4·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s − 1/3·9-s + 0.603·11-s + 2.77·13-s + 2.06·15-s + 1.45·17-s − 0.834·23-s + 2·25-s − 0.384·27-s − 0.371·29-s − 2.15·31-s + 0.696·33-s + 3.20·39-s + 1.87·41-s − 1.21·43-s − 0.596·45-s − 0.291·47-s + 1.68·51-s − 0.549·53-s + 1.07·55-s + 1.04·59-s + 2.56·61-s + 4.96·65-s − 0.977·67-s − 0.963·69-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.50012041\)
\(L(\frac12)\) \(\approx\) \(17.50012041\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 26 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 3 p T^{2} - 46 T^{3} + 480 T^{4} - 46 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + p T^{2} - 62 T^{3} + 434 T^{4} - 62 p T^{5} + p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} + 24 T^{3} + 1186 T^{4} + 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 54 T^{2} + 324 T^{3} + 1442 T^{4} + 324 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 134 T^{2} + 828 T^{3} + 5602 T^{4} + 828 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 106 T^{2} - 40 T^{3} + 5114 T^{4} - 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 68 T^{2} + 52 T^{3} - 2014 T^{4} + 52 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 124 T^{2} + 840 T^{3} + 7718 T^{4} + 840 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 147 T^{2} + 378 T^{3} + 9344 T^{4} + 378 p T^{5} + 147 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + 8866 T^{4} + 4 p^{2} T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 288 T^{2} - 2572 T^{3} + 22350 T^{4} - 2572 p T^{5} + 288 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 154 T^{2} + 1176 T^{3} + 13994 T^{4} + 1176 p T^{5} + 154 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 12 p T^{3} + 5438 T^{4} - 12 p^{2} T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 122 T^{2} + 240 T^{3} - 7038 T^{4} + 240 p T^{5} + 122 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 389 T^{2} - 4726 T^{3} + 48924 T^{4} - 4726 p T^{5} + 389 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 36 T + 750 T^{2} - 10564 T^{3} + 110978 T^{4} - 10564 p T^{5} + 750 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 914 T^{2} - 13800 T^{3} + 152258 T^{4} - 13800 p T^{5} + 914 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 545 T^{2} - 7602 T^{3} + 85890 T^{4} - 7602 p T^{5} + 545 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.59083465252474387026818317334, −6.21101113640725637933791823883, −6.09409783959105295914975673052, −5.93924520312052935722089767184, −5.80885466309380783405126181589, −5.44437155637316799194024938588, −5.33180710674637427631949292056, −5.08643217852243509721350705530, −5.07488861993010310787710489490, −4.46194808982191991615446706978, −4.42220528583697414976602865377, −3.88812825835473333307733620093, −3.79800485130032958593962780087, −3.40520815625714108343565622881, −3.39197008299875536973001812958, −3.31170357022331402214080482606, −3.23540282441840795076337610121, −2.25392677019954982852187135731, −2.24617518121900846534714467486, −2.22259092369611829856722206488, −2.12396932400775437920120258057, −1.41329511609365407501828709572, −1.30935426042833557702822533132, −0.790431648993298961563342424547, −0.73315088813029127302897487033, 0.73315088813029127302897487033, 0.790431648993298961563342424547, 1.30935426042833557702822533132, 1.41329511609365407501828709572, 2.12396932400775437920120258057, 2.22259092369611829856722206488, 2.24617518121900846534714467486, 2.25392677019954982852187135731, 3.23540282441840795076337610121, 3.31170357022331402214080482606, 3.39197008299875536973001812958, 3.40520815625714108343565622881, 3.79800485130032958593962780087, 3.88812825835473333307733620093, 4.42220528583697414976602865377, 4.46194808982191991615446706978, 5.07488861993010310787710489490, 5.08643217852243509721350705530, 5.33180710674637427631949292056, 5.44437155637316799194024938588, 5.80885466309380783405126181589, 5.93924520312052935722089767184, 6.09409783959105295914975673052, 6.21101113640725637933791823883, 6.59083465252474387026818317334

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