Properties

Label 8-370e4-1.1-c1e4-0-1
Degree $8$
Conductor $18741610000$
Sign $1$
Analytic cond. $76.1930$
Root an. cond. $1.71885$
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-s + 4·5-s − 6·9-s − 12·11-s − 6·19-s + 4·20-s + 5·25-s + 24·29-s − 16·31-s − 6·36-s + 20·41-s − 12·44-s − 24·45-s + 2·49-s − 48·55-s + 30·59-s − 24·61-s − 64-s + 12·71-s − 6·76-s + 8·79-s + 9·81-s − 30·89-s − 24·95-s + 72·99-s + 5·100-s − 16·101-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s − 2·9-s − 3.61·11-s − 1.37·19-s + 0.894·20-s + 25-s + 4.45·29-s − 2.87·31-s − 36-s + 3.12·41-s − 1.80·44-s − 3.57·45-s + 2/7·49-s − 6.47·55-s + 3.90·59-s − 3.07·61-s − 1/8·64-s + 1.42·71-s − 0.688·76-s + 0.900·79-s + 81-s − 3.17·89-s − 2.46·95-s + 7.23·99-s + 1/2·100-s − 1.59·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(76.1930\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{370} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.168600740\)
\(L(\frac12)\) \(\approx\) \(1.168600740\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^3$ \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 27 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$C_2^3$ \( 1 + 130 T^{2} + 10011 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.179697184492945270803829271806, −8.154707599900620657938405971372, −7.903714484017101141703843967704, −7.64574026710054425710151173162, −7.18836893378786406348373422968, −6.90944550808177087228869834061, −6.85941695458242684411104117143, −6.33406710812973675245333743579, −6.12911843667100302064604015565, −5.88272133762083988293051309975, −5.60962180821293832263610493412, −5.58251888620118908566387145747, −5.33810577372484577096069110447, −4.93361890103380899736607386423, −4.79364145498103143262749446962, −4.35823241665747837807536057036, −3.98976550893068370746307652244, −3.46208207938565518537077697797, −2.85172604290185480837619227158, −2.81772696693861262975472687071, −2.47065844549861953470922904189, −2.42756932472951083495926522505, −2.15608283953491981811341272537, −1.34375886735816440452922574959, −0.40222237310878924378328645053, 0.40222237310878924378328645053, 1.34375886735816440452922574959, 2.15608283953491981811341272537, 2.42756932472951083495926522505, 2.47065844549861953470922904189, 2.81772696693861262975472687071, 2.85172604290185480837619227158, 3.46208207938565518537077697797, 3.98976550893068370746307652244, 4.35823241665747837807536057036, 4.79364145498103143262749446962, 4.93361890103380899736607386423, 5.33810577372484577096069110447, 5.58251888620118908566387145747, 5.60962180821293832263610493412, 5.88272133762083988293051309975, 6.12911843667100302064604015565, 6.33406710812973675245333743579, 6.85941695458242684411104117143, 6.90944550808177087228869834061, 7.18836893378786406348373422968, 7.64574026710054425710151173162, 7.903714484017101141703843967704, 8.154707599900620657938405971372, 8.179697184492945270803829271806

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