Properties

Label 4-2475e2-1.1-c1e2-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·4-s + 7-s + 3·8-s − 2·11-s + 8·13-s − 14-s + 16-s − 17-s + 2·22-s − 3·23-s − 8·26-s − 2·28-s + 5·29-s − 6·31-s − 2·32-s + 34-s + 16·37-s + 6·41-s − 12·43-s + 4·44-s + 3·46-s − 6·47-s − 2·49-s − 16·52-s − 3·53-s + 3·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s + 0.377·7-s + 1.06·8-s − 0.603·11-s + 2.21·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.426·22-s − 0.625·23-s − 1.56·26-s − 0.377·28-s + 0.928·29-s − 1.07·31-s − 0.353·32-s + 0.171·34-s + 2.63·37-s + 0.937·41-s − 1.82·43-s + 0.603·44-s + 0.442·46-s − 0.875·47-s − 2/7·49-s − 2.21·52-s − 0.412·53-s + 0.400·56-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

\(L(1)\) \(\approx\) \(1.435097467\)
\(L(\frac12)\) \(\approx\) \(1.435097467\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 16 T + 133 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 23 T + 277 T^{2} - 23 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 5 T + 153 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 27 T + 337 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 25 T + 303 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T + 193 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−8.941659373707782851068029587110, −8.923641811610191952005488595345, −8.323491105795828004422478385486, −7.894215636803311273179184848065, −7.87291854586124570362650004911, −7.68866108884944240775195028373, −6.45817892164130181003547533687, −6.42618521545941063802048059409, −6.35320534499018699352081772823, −5.54783387358895014371530643234, −4.97635323305849684180720540501, −4.97427209371664079860068142091, −4.24430699203697829255624211409, −3.98800753877066793284880343359, −3.41074288090260044902666833996, −3.11298100691947915189548158533, −2.14779495820259948262395512900, −1.81765316420836059210672361330, −0.909372251615213281544084534352, −0.62962301905626252087556351454, 0.62962301905626252087556351454, 0.909372251615213281544084534352, 1.81765316420836059210672361330, 2.14779495820259948262395512900, 3.11298100691947915189548158533, 3.41074288090260044902666833996, 3.98800753877066793284880343359, 4.24430699203697829255624211409, 4.97427209371664079860068142091, 4.97635323305849684180720540501, 5.54783387358895014371530643234, 6.35320534499018699352081772823, 6.42618521545941063802048059409, 6.45817892164130181003547533687, 7.68866108884944240775195028373, 7.87291854586124570362650004911, 7.894215636803311273179184848065, 8.323491105795828004422478385486, 8.923641811610191952005488595345, 8.941659373707782851068029587110

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