Properties

Label 4-45e4-1.1-c1e2-0-10
Degree $4$
Conductor $4100625$
Sign $1$
Analytic cond. $261.459$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5·7-s − 8-s + 7·11-s − 3·13-s + 5·14-s − 16-s − 17-s − 7·22-s + 3·23-s + 3·26-s − 10·29-s + 2·31-s + 6·32-s + 34-s − 7·37-s − 7·41-s − 15·43-s − 3·46-s + 2·47-s + 8·49-s − 7·53-s + 5·56-s + 10·58-s − 11·59-s + 3·61-s − 2·62-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.88·7-s − 0.353·8-s + 2.11·11-s − 0.832·13-s + 1.33·14-s − 1/4·16-s − 0.242·17-s − 1.49·22-s + 0.625·23-s + 0.588·26-s − 1.85·29-s + 0.359·31-s + 1.06·32-s + 0.171·34-s − 1.15·37-s − 1.09·41-s − 2.28·43-s − 0.442·46-s + 0.291·47-s + 8/7·49-s − 0.961·53-s + 0.668·56-s + 1.31·58-s − 1.43·59-s + 0.384·61-s − 0.254·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(261.459\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4100625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 7 T + 91 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 11 T + 145 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 125 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 193 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 169 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.973065119240610218994664307948, −8.868690683377615967395287332808, −8.359310237906250390225410622537, −7.79927865824070916546171687645, −7.13839152537451634673526999456, −7.08902750059556208095645451281, −6.57599123035597623924556268568, −6.26356572111242214913721173472, −6.14256723438057370204199409255, −5.38964965320481471895425561888, −4.79875228104876865652184101698, −4.55124745589864639679683472009, −3.65893507099420483490640542397, −3.51988171783827466038401173268, −3.22176565712262229201201860183, −2.47783924235619733836826541904, −1.80646194354284746989280897111, −1.24047089183115338719865043265, 0, 0, 1.24047089183115338719865043265, 1.80646194354284746989280897111, 2.47783924235619733836826541904, 3.22176565712262229201201860183, 3.51988171783827466038401173268, 3.65893507099420483490640542397, 4.55124745589864639679683472009, 4.79875228104876865652184101698, 5.38964965320481471895425561888, 6.14256723438057370204199409255, 6.26356572111242214913721173472, 6.57599123035597623924556268568, 7.08902750059556208095645451281, 7.13839152537451634673526999456, 7.79927865824070916546171687645, 8.359310237906250390225410622537, 8.868690683377615967395287332808, 8.973065119240610218994664307948

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