Properties

Label 2-588-1.1-c7-0-45
Degree $2$
Conductor $588$
Sign $-1$
Analytic cond. $183.682$
Root an. cond. $13.5529$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s + 378·5-s + 729·9-s − 2.48e3·11-s − 1.48e4·13-s + 1.02e4·15-s + 2.23e4·17-s + 1.63e4·19-s − 1.15e5·23-s + 6.47e4·25-s + 1.96e4·27-s + 1.57e5·29-s + 1.64e4·31-s − 6.70e4·33-s − 1.49e5·37-s − 4.01e5·39-s + 2.41e5·41-s − 4.43e5·43-s + 2.75e5·45-s − 9.22e5·47-s + 6.02e5·51-s − 6.97e5·53-s − 9.38e5·55-s + 4.40e5·57-s − 8.70e5·59-s − 2.06e6·61-s − 5.62e6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.35·5-s + 1/3·9-s − 0.562·11-s − 1.87·13-s + 0.780·15-s + 1.10·17-s + 0.545·19-s − 1.97·23-s + 0.828·25-s + 0.192·27-s + 1.19·29-s + 0.0992·31-s − 0.324·33-s − 0.484·37-s − 1.08·39-s + 0.546·41-s − 0.850·43-s + 0.450·45-s − 1.29·47-s + 0.635·51-s − 0.643·53-s − 0.760·55-s + 0.314·57-s − 0.551·59-s − 1.16·61-s − 2.53·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(183.682\)
Root analytic conductor: \(13.5529\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 588,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p^{3} T \)
7 \( 1 \)
good5 \( 1 - 378 T + p^{7} T^{2} \)
11 \( 1 + 2484 T + p^{7} T^{2} \)
13 \( 1 + 14870 T + p^{7} T^{2} \)
17 \( 1 - 22302 T + p^{7} T^{2} \)
19 \( 1 - 16300 T + p^{7} T^{2} \)
23 \( 1 + 115128 T + p^{7} T^{2} \)
29 \( 1 - 157086 T + p^{7} T^{2} \)
31 \( 1 - 16456 T + p^{7} T^{2} \)
37 \( 1 + 149266 T + p^{7} T^{2} \)
41 \( 1 - 241110 T + p^{7} T^{2} \)
43 \( 1 + 443188 T + p^{7} T^{2} \)
47 \( 1 + 922752 T + p^{7} T^{2} \)
53 \( 1 + 697626 T + p^{7} T^{2} \)
59 \( 1 + 870156 T + p^{7} T^{2} \)
61 \( 1 + 2067062 T + p^{7} T^{2} \)
67 \( 1 + 1680748 T + p^{7} T^{2} \)
71 \( 1 + 1070280 T + p^{7} T^{2} \)
73 \( 1 - 2403334 T + p^{7} T^{2} \)
79 \( 1 - 2301512 T + p^{7} T^{2} \)
83 \( 1 + 4708692 T + p^{7} T^{2} \)
89 \( 1 + 4143690 T + p^{7} T^{2} \)
97 \( 1 - 1622974 T + p^{7} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.441025784200162792123868338129, −8.155529390396923947603947844678, −7.51185192925673372658253498523, −6.37325683729545528767995653274, −5.44573072716019021701176113949, −4.65595581920032062720951711162, −3.13881031276499777922969470670, −2.34517574691463211133287335349, −1.51494570046796892746188961605, 0, 1.51494570046796892746188961605, 2.34517574691463211133287335349, 3.13881031276499777922969470670, 4.65595581920032062720951711162, 5.44573072716019021701176113949, 6.37325683729545528767995653274, 7.51185192925673372658253498523, 8.155529390396923947603947844678, 9.441025784200162792123868338129

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