Properties

Label 4-3525e2-1.1-c1e2-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 3·8-s + 3·9-s + 7·11-s + 2·12-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s + 12·19-s − 2·21-s + 7·22-s + 3·23-s + 6·24-s + 6·26-s + 4·27-s − 28-s − 15·29-s + 6·31-s − 32-s + 14·33-s − 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 1.06·8-s + 9-s + 2.11·11-s + 0.577·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 2.75·19-s − 0.436·21-s + 1.49·22-s + 0.625·23-s + 1.22·24-s + 1.17·26-s + 0.769·27-s − 0.188·28-s − 2.78·29-s + 1.07·31-s − 0.176·32-s + 2.43·33-s − 0.342·34-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

\(L(1)\) \(\approx\) \(11.31938477\)
\(L(\frac12)\) \(\approx\) \(11.31938477\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 15 T + 110 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 15 T + 210 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
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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

−8.619839010287527726123048282061, −8.532021386644083971162390704030, −8.125885562189757696159443134353, −7.32494797928465343466788187974, −7.21337777476355516420931328990, −7.10577924616050045406455547789, −6.65399523212878187514296852989, −6.12231734038658971106213126440, −5.72564681948181904840733328703, −5.45581911451406974326889384389, −4.69152801212127144907335384724, −4.54844311995279887696006024032, −3.95765226264758390153285765843, −3.46829556796645367684705152729, −3.37489841745021752124814137040, −3.24310939014855361679617182597, −2.22820754835305245913204896375, −1.58207226202124541382049686365, −1.56826269906311933645701924599, −0.882930258623306524822922614315, 0.882930258623306524822922614315, 1.56826269906311933645701924599, 1.58207226202124541382049686365, 2.22820754835305245913204896375, 3.24310939014855361679617182597, 3.37489841745021752124814137040, 3.46829556796645367684705152729, 3.95765226264758390153285765843, 4.54844311995279887696006024032, 4.69152801212127144907335384724, 5.45581911451406974326889384389, 5.72564681948181904840733328703, 6.12231734038658971106213126440, 6.65399523212878187514296852989, 7.10577924616050045406455547789, 7.21337777476355516420931328990, 7.32494797928465343466788187974, 8.125885562189757696159443134353, 8.532021386644083971162390704030, 8.619839010287527726123048282061

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