Properties

Label 4-2475e2-1.1-c1e2-0-4
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
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  + 3·4-s + 2·11-s + 5·16-s + 2·19-s − 12·29-s + 8·31-s − 10·41-s + 6·44-s + 5·49-s − 22·59-s + 28·61-s + 3·64-s − 10·71-s + 6·76-s − 10·79-s + 20·89-s − 22·101-s + 24·109-s − 36·116-s + 3·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.603·11-s + 5/4·16-s + 0.458·19-s − 2.22·29-s + 1.43·31-s − 1.56·41-s + 0.904·44-s + 5/7·49-s − 2.86·59-s + 3.58·61-s + 3/8·64-s − 1.18·71-s + 0.688·76-s − 1.12·79-s + 2.11·89-s − 2.18·101-s + 2.29·109-s − 3.34·116-s + 3/11·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.764312192\)
\(L(\frac12)\) \(\approx\) \(3.764312192\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−9.023805898248617786382142580890, −8.699408782918959626345049037874, −8.421432771608845102549341647750, −7.72394666564413383604960560338, −7.58295097358687431707525205060, −7.22270835912240985655539634680, −6.82065039346092720690390776456, −6.40720881365819421235420583893, −6.21206327067784175254319894960, −5.58248858524698529005924088828, −5.44057641044720505529281686741, −4.75860537385674775074185387345, −4.34226520463857271465691425343, −3.58166234737115796904281772330, −3.55221791758396881272734347657, −2.80141425953824549116473130474, −2.47523340911041945920181983125, −1.67935290940881962576708987127, −1.61714311120405321001516349843, −0.61942529144922576853363480471, 0.61942529144922576853363480471, 1.61714311120405321001516349843, 1.67935290940881962576708987127, 2.47523340911041945920181983125, 2.80141425953824549116473130474, 3.55221791758396881272734347657, 3.58166234737115796904281772330, 4.34226520463857271465691425343, 4.75860537385674775074185387345, 5.44057641044720505529281686741, 5.58248858524698529005924088828, 6.21206327067784175254319894960, 6.40720881365819421235420583893, 6.82065039346092720690390776456, 7.22270835912240985655539634680, 7.58295097358687431707525205060, 7.72394666564413383604960560338, 8.421432771608845102549341647750, 8.699408782918959626345049037874, 9.023805898248617786382142580890

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