Properties

Label 4-2475e2-1.1-c1e2-0-12
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-s + 4-s − 2·7-s + 3·8-s + 2·11-s + 5·13-s − 2·14-s + 16-s + 7·17-s + 6·19-s + 2·22-s + 9·23-s + 5·26-s − 2·28-s − 2·29-s − 3·31-s − 32-s + 7·34-s + 37-s + 6·38-s + 3·41-s + 3·43-s + 2·44-s + 9·46-s + 15·47-s − 11·49-s + 5·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 1.06·8-s + 0.603·11-s + 1.38·13-s − 0.534·14-s + 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.426·22-s + 1.87·23-s + 0.980·26-s − 0.377·28-s − 0.371·29-s − 0.538·31-s − 0.176·32-s + 1.20·34-s + 0.164·37-s + 0.973·38-s + 0.468·41-s + 0.457·43-s + 0.301·44-s + 1.32·46-s + 2.18·47-s − 1.57·49-s + 0.693·52-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

\(L(1)\) \(\approx\) \(5.833669291\)
\(L(\frac12)\) \(\approx\) \(5.833669291\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_4$ \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 96 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 19 T + 210 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_4$ \( 1 - 26 T + 330 T^{2} - 26 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 181 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−9.027994156606032544069874377355, −9.008353773601482822666384632232, −8.317427360164500719333675615895, −7.73340111824441623790719979833, −7.53549529188499791817938419355, −7.18263939899137526638170993091, −6.91381696389274490866972809032, −6.26510694885346203851524678458, −5.89337302097861147499773196989, −5.79500240746988737288128213927, −5.12127610624148283958922911689, −4.86949119882560381764474549798, −4.21525007864685896073238498892, −3.91690925270182544016244310188, −3.25117196511049350532660073447, −3.21110396665105094870106843254, −2.70762100408032687538616862079, −1.68082303157079390627616932394, −1.33334055688953324011476936597, −0.810047830414422836938202944158, 0.810047830414422836938202944158, 1.33334055688953324011476936597, 1.68082303157079390627616932394, 2.70762100408032687538616862079, 3.21110396665105094870106843254, 3.25117196511049350532660073447, 3.91690925270182544016244310188, 4.21525007864685896073238498892, 4.86949119882560381764474549798, 5.12127610624148283958922911689, 5.79500240746988737288128213927, 5.89337302097861147499773196989, 6.26510694885346203851524678458, 6.91381696389274490866972809032, 7.18263939899137526638170993091, 7.53549529188499791817938419355, 7.73340111824441623790719979833, 8.317427360164500719333675615895, 9.008353773601482822666384632232, 9.027994156606032544069874377355

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