Properties

Label 4-75e4-1.1-c1e2-0-5
Degree $4$
Conductor $31640625$
Sign $1$
Analytic cond. $2017.43$
Root an. cond. $6.70192$
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  + 2-s − 2·4-s + 4·7-s − 3·8-s + 6·11-s + 2·13-s + 4·14-s + 16-s − 4·17-s + 6·22-s + 13·23-s + 2·26-s − 8·28-s + 5·29-s + 4·31-s + 2·32-s − 4·34-s + 4·37-s + 21·41-s + 17·43-s − 12·44-s + 13·46-s − 14·47-s − 2·49-s − 4·52-s + 18·53-s − 12·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 1.51·7-s − 1.06·8-s + 1.80·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 1.27·22-s + 2.71·23-s + 0.392·26-s − 1.51·28-s + 0.928·29-s + 0.718·31-s + 0.353·32-s − 0.685·34-s + 0.657·37-s + 3.27·41-s + 2.59·43-s − 1.80·44-s + 1.91·46-s − 2.04·47-s − 2/7·49-s − 0.554·52-s + 2.47·53-s − 1.60·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2017.43\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 31640625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.512568788\)
\(L(\frac12)\) \(\approx\) \(6.512568788\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_4$ \( 1 - 21 T + 191 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 17 T + 157 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−8.296718245912122637367585276106, −8.193882336294560867940576946385, −7.51193252831418462739839390973, −7.16517218523111685404887345631, −6.98607920434936716189001580065, −6.42937178358094565300146826854, −6.04315260952597940359726895089, −5.83080796511311859962397435883, −5.22732470707739054699616299345, −5.00335411200033079020636110645, −4.47877354462660298716623665938, −4.38291838802353800698481200700, −4.07030818396533060539590693767, −3.80172715597284945757639283753, −2.90955978506035313830731761930, −2.84856787953972731169411469168, −2.16523648287972823866689000080, −1.44609774381445406962922434299, −0.892707202049531009233107064178, −0.874120518358492336014150497013, 0.874120518358492336014150497013, 0.892707202049531009233107064178, 1.44609774381445406962922434299, 2.16523648287972823866689000080, 2.84856787953972731169411469168, 2.90955978506035313830731761930, 3.80172715597284945757639283753, 4.07030818396533060539590693767, 4.38291838802353800698481200700, 4.47877354462660298716623665938, 5.00335411200033079020636110645, 5.22732470707739054699616299345, 5.83080796511311859962397435883, 6.04315260952597940359726895089, 6.42937178358094565300146826854, 6.98607920434936716189001580065, 7.16517218523111685404887345631, 7.51193252831418462739839390973, 8.193882336294560867940576946385, 8.296718245912122637367585276106

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