Properties

Label 4-3525e2-1.1-c1e2-0-0
Degree $4$
Conductor $12425625$
Sign $1$
Analytic cond. $792.268$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 4-s − 4·6-s + 6·7-s + 3·9-s − 2·12-s + 10·13-s + 12·14-s + 16-s + 6·17-s + 6·18-s − 6·19-s − 12·21-s + 6·23-s + 20·26-s − 4·27-s + 6·28-s − 6·29-s + 4·31-s − 2·32-s + 12·34-s + 3·36-s + 4·37-s − 12·38-s − 20·39-s + 2·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s + 2.26·7-s + 9-s − 0.577·12-s + 2.77·13-s + 3.20·14-s + 1/4·16-s + 1.45·17-s + 1.41·18-s − 1.37·19-s − 2.61·21-s + 1.25·23-s + 3.92·26-s − 0.769·27-s + 1.13·28-s − 1.11·29-s + 0.718·31-s − 0.353·32-s + 2.05·34-s + 1/2·36-s + 0.657·37-s − 1.94·38-s − 3.20·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12425625\)    =    \(3^{2} \cdot 5^{4} \cdot 47^{2}\)
Sign: $1$
Analytic conductor: \(792.268\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12425625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.300025568\)
\(L(\frac12)\) \(\approx\) \(7.300025568\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
47$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 10 T + 49 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 141 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 4 T - 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 18 T + 205 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.593406178457278095582564003420, −8.316234623504708131738551627881, −8.034642318780751431940544473276, −7.49173197974669354120288113854, −7.36977414382382955234155728902, −6.69654994411818250381677239807, −6.12324942351913695913053388822, −6.09209136024998791608340193702, −5.56744817428610029293446090707, −5.39968792332398030484390655796, −4.93298949538984324643966627615, −4.56132388016260890730650463516, −4.21387095470995261423925390520, −4.07713477221879794941271149129, −3.40782902540093234419563600956, −3.13674717328842296757083875478, −2.12641598374788445386872401865, −1.59587759441434326138288826046, −1.27420519025417126204684263572, −0.800168571793177916973202622197, 0.800168571793177916973202622197, 1.27420519025417126204684263572, 1.59587759441434326138288826046, 2.12641598374788445386872401865, 3.13674717328842296757083875478, 3.40782902540093234419563600956, 4.07713477221879794941271149129, 4.21387095470995261423925390520, 4.56132388016260890730650463516, 4.93298949538984324643966627615, 5.39968792332398030484390655796, 5.56744817428610029293446090707, 6.09209136024998791608340193702, 6.12324942351913695913053388822, 6.69654994411818250381677239807, 7.36977414382382955234155728902, 7.49173197974669354120288113854, 8.034642318780751431940544473276, 8.316234623504708131738551627881, 8.593406178457278095582564003420

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