Properties

Degree $4$
Conductor $63504$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 2·11-s − 6·13-s + 8·17-s + 19-s + 8·23-s + 5·25-s − 8·29-s − 3·31-s − 2·35-s + 37-s − 12·41-s + 22·43-s + 6·47-s − 6·49-s − 12·53-s − 4·55-s + 4·59-s + 6·61-s + 12·65-s − 13·67-s + 20·71-s + 11·73-s + 2·77-s + 3·79-s − 4·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.603·11-s − 1.66·13-s + 1.94·17-s + 0.229·19-s + 1.66·23-s + 25-s − 1.48·29-s − 0.538·31-s − 0.338·35-s + 0.164·37-s − 1.87·41-s + 3.35·43-s + 0.875·47-s − 6/7·49-s − 1.64·53-s − 0.539·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 1.58·67-s + 2.37·71-s + 1.28·73-s + 0.227·77-s + 0.337·79-s − 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63504,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26315\)
\(L(\frac12)\) \(\approx\) \(1.26315\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−12.31452891717860794310765759901, −11.83503703506292502175025376957, −11.28791527001990603061071035990, −11.04531449514874741828729448281, −10.42924890499627037924724190065, −9.812583697983873621121745053008, −9.372159469906026931154183920467, −9.042130284714622882543932201253, −8.299391711674754658549173330954, −7.71444404342075596320824127988, −7.30990981320539886578536057919, −7.19961418344622387872940235902, −6.25554651021895590681373542408, −5.48305515935114243213300889725, −5.04613988998220734418384382401, −4.54400292622656144790669268154, −3.60203331204938496924430643433, −3.25553039821128219080739947707, −2.23241068812336342992105072826, −0.965795568859102392831399663720, 0.965795568859102392831399663720, 2.23241068812336342992105072826, 3.25553039821128219080739947707, 3.60203331204938496924430643433, 4.54400292622656144790669268154, 5.04613988998220734418384382401, 5.48305515935114243213300889725, 6.25554651021895590681373542408, 7.19961418344622387872940235902, 7.30990981320539886578536057919, 7.71444404342075596320824127988, 8.299391711674754658549173330954, 9.042130284714622882543932201253, 9.372159469906026931154183920467, 9.812583697983873621121745053008, 10.42924890499627037924724190065, 11.04531449514874741828729448281, 11.28791527001990603061071035990, 11.83503703506292502175025376957, 12.31452891717860794310765759901

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