Properties

Label 4-8470e2-1.1-c1e2-0-10
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 + 3·4-s − 2·5-s + 2·7-s − 4·8-s + 2·9-s + 4·10-s − 4·13-s − 4·14-s + 5·16-s − 4·17-s − 4·18-s − 6·20-s + 3·25-s + 8·26-s + 6·28-s − 12·29-s − 6·32-s + 8·34-s − 4·35-s + 6·36-s + 12·37-s + 8·40-s − 4·41-s + 8·43-s − 4·45-s − 16·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.970·17-s − 0.942·18-s − 1.34·20-s + 3/5·25-s + 1.56·26-s + 1.13·28-s − 2.22·29-s − 1.06·32-s + 1.37·34-s − 0.676·35-s + 36-s + 1.97·37-s + 1.26·40-s − 0.624·41-s + 1.21·43-s − 0.596·45-s − 2.33·47-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 214 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 190 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.69290706061046991419729170522, −7.64985001706633197573353688033, −7.12410196158094950650297801529, −6.62923271545596838305663194834, −6.41715937623756269132467792109, −6.23735449884101537305858375725, −5.36737590160804753915907662625, −5.27176126435502999331733538728, −4.86959097352514473556504623510, −4.33114404947085658146874566075, −4.07166116468635060645889135818, −3.78682559478378396781756919967, −3.01740286926770876689505843721, −2.82989658276470805211087713559, −2.25843032292322031240915289699, −1.82238133592809602959791585716, −1.50404889698204852610377958237, −0.913297055630509466733777954132, 0, 0, 0.913297055630509466733777954132, 1.50404889698204852610377958237, 1.82238133592809602959791585716, 2.25843032292322031240915289699, 2.82989658276470805211087713559, 3.01740286926770876689505843721, 3.78682559478378396781756919967, 4.07166116468635060645889135818, 4.33114404947085658146874566075, 4.86959097352514473556504623510, 5.27176126435502999331733538728, 5.36737590160804753915907662625, 6.23735449884101537305858375725, 6.41715937623756269132467792109, 6.62923271545596838305663194834, 7.12410196158094950650297801529, 7.64985001706633197573353688033, 7.69290706061046991419729170522

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