Properties

Label 4-5054e2-1.1-c1e2-0-2
Degree $4$
Conductor $25542916$
Sign $1$
Analytic cond. $1628.63$
Root an. cond. $6.35266$
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  − 2·2-s + 3-s + 3·4-s − 3·5-s − 2·6-s + 2·7-s − 4·8-s − 4·9-s + 6·10-s − 3·11-s + 3·12-s − 4·14-s − 3·15-s + 5·16-s + 6·17-s + 8·18-s − 9·20-s + 2·21-s + 6·22-s − 4·24-s − 2·25-s − 6·27-s + 6·28-s + 7·29-s + 6·30-s + 12·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 1.34·5-s − 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.89·10-s − 0.904·11-s + 0.866·12-s − 1.06·14-s − 0.774·15-s + 5/4·16-s + 1.45·17-s + 1.88·18-s − 2.01·20-s + 0.436·21-s + 1.27·22-s − 0.816·24-s − 2/5·25-s − 1.15·27-s + 1.13·28-s + 1.29·29-s + 1.09·30-s + 2.15·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25542916\)    =    \(2^{2} \cdot 7^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1628.63\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5054} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25542916,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.136745465\)
\(L(\frac12)\) \(\approx\) \(1.136745465\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
19 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 59 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T - 19 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 75 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 17 T + 167 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 15 T + 153 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 13 T + 209 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
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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

−8.188036296471928122640384861596, −8.162646111900772498179998260656, −7.88725589967397767720685208378, −7.61141231546193969286285753974, −7.40085483958602519693866553164, −6.73566626773931879809450467347, −6.27883078710622879192025017049, −5.99273230443321238428187674388, −5.72674116193616171270915631150, −5.04646806608112318358805159491, −4.66859573531801999687659414156, −4.47951579912647011050824795580, −3.67164167160980973347791700435, −3.27135575430449795238275921617, −2.93208619565584855613044444588, −2.78165750864897515899034714034, −2.04147363234434574697612834699, −1.62953851181612196384905410238, −0.71292337534387395828772496703, −0.52957002926456084968500642065, 0.52957002926456084968500642065, 0.71292337534387395828772496703, 1.62953851181612196384905410238, 2.04147363234434574697612834699, 2.78165750864897515899034714034, 2.93208619565584855613044444588, 3.27135575430449795238275921617, 3.67164167160980973347791700435, 4.47951579912647011050824795580, 4.66859573531801999687659414156, 5.04646806608112318358805159491, 5.72674116193616171270915631150, 5.99273230443321238428187674388, 6.27883078710622879192025017049, 6.73566626773931879809450467347, 7.40085483958602519693866553164, 7.61141231546193969286285753974, 7.88725589967397767720685208378, 8.162646111900772498179998260656, 8.188036296471928122640384861596

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