Properties

Label 4-2475e2-1.1-c1e2-0-11
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  + 2·2-s + 4-s + 2·7-s + 2·11-s + 4·14-s + 16-s + 2·17-s + 10·19-s + 4·22-s + 2·23-s + 2·28-s − 8·29-s − 2·32-s + 4·34-s + 6·37-s + 20·38-s + 2·41-s + 12·43-s + 2·44-s + 4·46-s + 2·47-s − 9·49-s + 4·53-s − 16·58-s − 22·59-s + 12·61-s − 11·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.755·7-s + 0.603·11-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 2.29·19-s + 0.852·22-s + 0.417·23-s + 0.377·28-s − 1.48·29-s − 0.353·32-s + 0.685·34-s + 0.986·37-s + 3.24·38-s + 0.312·41-s + 1.82·43-s + 0.301·44-s + 0.589·46-s + 0.291·47-s − 9/7·49-s + 0.549·53-s − 2.10·58-s − 2.86·59-s + 1.53·61-s − 1.37·64-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

\(L(1)\) \(\approx\) \(7.197432990\)
\(L(\frac12)\) \(\approx\) \(7.197432990\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 159 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−9.042108873220691970721145327688, −9.038303246615674129741912420181, −8.047789792445424609641471308052, −7.78903571165956974370921895178, −7.74193408000098379774415786049, −7.21133434994057695213254037416, −6.74719232777622347283862389862, −6.28788355886798183937497448502, −5.67279215383912692861881256720, −5.60139774587708668935289470370, −5.05311411854726103041484059790, −4.91449834580153901885306396802, −4.30585944351200599425917937624, −3.98889211675823804481276091346, −3.47813171042685847801851810315, −3.25409295182062201412747301077, −2.58177776637967524473544864204, −1.94093501066343588614189941391, −1.29914144667592596561329551750, −0.798073002033207128150251416184, 0.798073002033207128150251416184, 1.29914144667592596561329551750, 1.94093501066343588614189941391, 2.58177776637967524473544864204, 3.25409295182062201412747301077, 3.47813171042685847801851810315, 3.98889211675823804481276091346, 4.30585944351200599425917937624, 4.91449834580153901885306396802, 5.05311411854726103041484059790, 5.60139774587708668935289470370, 5.67279215383912692861881256720, 6.28788355886798183937497448502, 6.74719232777622347283862389862, 7.21133434994057695213254037416, 7.74193408000098379774415786049, 7.78903571165956974370921895178, 8.047789792445424609641471308052, 9.038303246615674129741912420181, 9.042108873220691970721145327688

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