Properties

Label 8-930e4-1.1-c1e4-0-3
Degree $8$
Conductor $748052010000$
Sign $1$
Analytic cond. $3041.16$
Root an. cond. $2.72508$
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  + 4·3-s + 8·9-s − 12·13-s − 16-s + 8·25-s + 12·27-s − 4·31-s − 20·37-s − 48·39-s + 32·43-s − 4·48-s − 32·61-s − 4·67-s + 8·73-s + 32·75-s + 23·81-s − 16·93-s + 12·97-s + 40·103-s − 80·111-s − 96·117-s + 8·121-s + 127-s + 128·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2.30·3-s + 8/3·9-s − 3.32·13-s − 1/4·16-s + 8/5·25-s + 2.30·27-s − 0.718·31-s − 3.28·37-s − 7.68·39-s + 4.87·43-s − 0.577·48-s − 4.09·61-s − 0.488·67-s + 0.936·73-s + 3.69·75-s + 23/9·81-s − 1.65·93-s + 1.21·97-s + 3.94·103-s − 7.59·111-s − 8.87·117-s + 8/11·121-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(3041.16\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.539086211\)
\(L(\frac12)\) \(\approx\) \(3.539086211\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_1$ \( ( 1 + T )^{4} \)
good7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 1054 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 718 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
89$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.41384087811808677296044834966, −7.11733374346453031684230475801, −7.08232546690759435107189515270, −6.65067766531095354132003118116, −6.24219707712050805445023914349, −6.20193679202515745442359201255, −6.02703724300821650857947394985, −5.43590190153375436318456036634, −5.15497109352716466242633866706, −5.01503037967307435874493270303, −4.98184243954086951435181798117, −4.56650918508809601958237593532, −4.44082368979591677309874429857, −3.94653290967398345098659463652, −3.89076450321693730876667080315, −3.50464013593630016629411799984, −3.10199337933578825288728933139, −3.02860342749846366064179571852, −2.62623386413265361043416781078, −2.51449263104060758167431677344, −2.34924541118969268871658147381, −1.78449513312604381026496152832, −1.77191542487634168713594702616, −1.03492709954164932533007146838, −0.35845225930331716462255307627, 0.35845225930331716462255307627, 1.03492709954164932533007146838, 1.77191542487634168713594702616, 1.78449513312604381026496152832, 2.34924541118969268871658147381, 2.51449263104060758167431677344, 2.62623386413265361043416781078, 3.02860342749846366064179571852, 3.10199337933578825288728933139, 3.50464013593630016629411799984, 3.89076450321693730876667080315, 3.94653290967398345098659463652, 4.44082368979591677309874429857, 4.56650918508809601958237593532, 4.98184243954086951435181798117, 5.01503037967307435874493270303, 5.15497109352716466242633866706, 5.43590190153375436318456036634, 6.02703724300821650857947394985, 6.20193679202515745442359201255, 6.24219707712050805445023914349, 6.65067766531095354132003118116, 7.08232546690759435107189515270, 7.11733374346453031684230475801, 7.41384087811808677296044834966

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