Properties

Label 4-1575e2-1.1-c1e2-0-3
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
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 − 2·7-s + 8-s + 6·11-s − 2·13-s − 2·14-s − 16-s + 2·17-s + 6·19-s + 6·22-s + 2·23-s − 2·26-s + 8·29-s − 6·32-s + 2·34-s − 6·37-s + 6·38-s − 12·43-s + 2·46-s + 14·47-s + 3·49-s − 8·53-s − 2·56-s + 8·58-s + 14·59-s + 12·61-s − 3·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.755·7-s + 0.353·8-s + 1.80·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 1.37·19-s + 1.27·22-s + 0.417·23-s − 0.392·26-s + 1.48·29-s − 1.06·32-s + 0.342·34-s − 0.986·37-s + 0.973·38-s − 1.82·43-s + 0.294·46-s + 2.04·47-s + 3/7·49-s − 1.09·53-s − 0.267·56-s + 1.05·58-s + 1.82·59-s + 1.53·61-s − 3/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.575642407\)
\(L(\frac12)\) \(\approx\) \(3.575642407\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 24 T + 265 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 206 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−9.555396062243884710584113877616, −9.277948927026265086072843465265, −8.877592668520726893469637519809, −8.582475244357688414558009217930, −7.977364365762281703477147644566, −7.49633896663876218492260612188, −7.04369091863314510300183569764, −6.73013673643529782886447792727, −6.58590204619360398955536186077, −5.88417886213956642443840378598, −5.41799338534272339725064127285, −5.12969305469015921393156494293, −4.52310423940609413112861753694, −4.22819737421636746451446292254, −3.61660217160819773398942478822, −3.34301704978670366630099398848, −2.82795638307420451373974857874, −2.05982445638797273093071187374, −1.37364258847294269654492510263, −0.71776872921996410606111884741, 0.71776872921996410606111884741, 1.37364258847294269654492510263, 2.05982445638797273093071187374, 2.82795638307420451373974857874, 3.34301704978670366630099398848, 3.61660217160819773398942478822, 4.22819737421636746451446292254, 4.52310423940609413112861753694, 5.12969305469015921393156494293, 5.41799338534272339725064127285, 5.88417886213956642443840378598, 6.58590204619360398955536186077, 6.73013673643529782886447792727, 7.04369091863314510300183569764, 7.49633896663876218492260612188, 7.977364365762281703477147644566, 8.582475244357688414558009217930, 8.877592668520726893469637519809, 9.277948927026265086072843465265, 9.555396062243884710584113877616

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