Properties

Label 4-8470e2-1.1-c1e2-0-20
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s + 4·8-s + 2·9-s + 4·10-s − 6·12-s − 4·14-s − 4·15-s + 5·16-s + 4·17-s + 4·18-s − 6·19-s + 6·20-s + 4·21-s − 10·23-s − 8·24-s + 3·25-s − 6·27-s − 6·28-s + 2·29-s − 8·30-s + 6·32-s + 8·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 0.942·18-s − 1.37·19-s + 1.34·20-s + 0.872·21-s − 2.08·23-s − 1.63·24-s + 3/5·25-s − 1.15·27-s − 1.13·28-s + 0.371·29-s − 1.46·30-s + 1.06·32-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 20 T + 202 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 162 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_4$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.52288990634885519651494560292, −6.92996313185405650257546706589, −6.65203275765255457412480435970, −6.44692416171195606953907312297, −6.03880807990291479605966724429, −6.01780895557364232237614387181, −5.54339967325523741573468428728, −5.32203172287614803980305899941, −4.73202350622291036944373109459, −4.68559827966020112781740588113, −4.13744539748480257134068059607, −3.71700680256760167583433578255, −3.35914745925531593028311679085, −3.07597485175026265559249709556, −2.42689688680661058357772083458, −2.00543099810267048338213191069, −1.66693095260151255300775971567, −1.22832229666786241891456387085, 0, 0, 1.22832229666786241891456387085, 1.66693095260151255300775971567, 2.00543099810267048338213191069, 2.42689688680661058357772083458, 3.07597485175026265559249709556, 3.35914745925531593028311679085, 3.71700680256760167583433578255, 4.13744539748480257134068059607, 4.68559827966020112781740588113, 4.73202350622291036944373109459, 5.32203172287614803980305899941, 5.54339967325523741573468428728, 6.01780895557364232237614387181, 6.03880807990291479605966724429, 6.44692416171195606953907312297, 6.65203275765255457412480435970, 6.92996313185405650257546706589, 7.52288990634885519651494560292

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