Properties

Label 4-525e2-1.1-c3e2-0-13
Degree $4$
Conductor $275625$
Sign $1$
Analytic cond. $959.512$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·3-s − 5·4-s − 18·6-s + 14·7-s + 33·8-s + 27·9-s − 31·11-s − 30·12-s − 39·13-s − 42·14-s − 21·16-s + 79·17-s − 81·18-s − 56·19-s + 84·21-s + 93·22-s − 254·23-s + 198·24-s + 117·26-s + 108·27-s − 70·28-s − 62·29-s − 135·31-s − 87·32-s − 186·33-s − 237·34-s + ⋯
L(s)  = 1  − 1.06·2-s + 1.15·3-s − 5/8·4-s − 1.22·6-s + 0.755·7-s + 1.45·8-s + 9-s − 0.849·11-s − 0.721·12-s − 0.832·13-s − 0.801·14-s − 0.328·16-s + 1.12·17-s − 1.06·18-s − 0.676·19-s + 0.872·21-s + 0.901·22-s − 2.30·23-s + 1.68·24-s + 0.882·26-s + 0.769·27-s − 0.472·28-s − 0.397·29-s − 0.782·31-s − 0.480·32-s − 0.981·33-s − 1.19·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(959.512\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 275625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good2$C_4$ \( 1 + 3 T + 7 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 3 p T + 3546 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 79 T + 8288 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 254 T + 38015 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 62 T + 19751 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 113 T + 102964 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 804 T + 306321 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 152 T + 180510 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 149 T + 269640 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 441 T + 273734 T^{2} + 441 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1157 T + 871890 T^{2} + 1157 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 268 T + 763078 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1211 T + 720720 T^{2} + 1211 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 466 T + 306170 T^{2} - 466 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 172 T - 273626 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.978844281186729085710753821351, −9.849708628900840625392888038927, −9.182752734382268522602132133826, −9.015677661701696983659479578508, −8.241889097431895984297069870114, −8.189074437725777677509591503270, −7.69531193756662675881230856221, −7.63069371584256306352029705758, −6.80038337320182259248395398509, −6.12807000959301484725777218404, −5.24431720393011742867523080898, −5.16679298048263652568632294577, −4.17748629582954838913062794466, −4.16619229371580336296790025746, −3.23530014965051729462661896334, −2.61924209101489875657644639319, −1.73211390683513546126124327273, −1.50634273801803250279664283532, 0, 0, 1.50634273801803250279664283532, 1.73211390683513546126124327273, 2.61924209101489875657644639319, 3.23530014965051729462661896334, 4.16619229371580336296790025746, 4.17748629582954838913062794466, 5.16679298048263652568632294577, 5.24431720393011742867523080898, 6.12807000959301484725777218404, 6.80038337320182259248395398509, 7.63069371584256306352029705758, 7.69531193756662675881230856221, 8.189074437725777677509591503270, 8.241889097431895984297069870114, 9.015677661701696983659479578508, 9.182752734382268522602132133826, 9.849708628900840625392888038927, 9.978844281186729085710753821351

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