Properties

Label 2-588-1.1-c3-0-14
Degree $2$
Conductor $588$
Sign $-1$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 9.82·5-s + 9·9-s − 14.1·11-s − 26.1·13-s − 29.4·15-s − 78.5·17-s + 73.1·19-s − 96·23-s − 28.4·25-s − 27·27-s + 173.·29-s + 67.2·31-s + 42.5·33-s − 301.·37-s + 78.3·39-s + 472.·41-s − 463.·43-s + 88.4·45-s − 91.1·47-s + 235.·51-s − 163.·53-s − 139.·55-s − 219.·57-s − 600.·59-s + 571.·61-s − 256.·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.878·5-s + 0.333·9-s − 0.388·11-s − 0.557·13-s − 0.507·15-s − 1.12·17-s + 0.883·19-s − 0.870·23-s − 0.227·25-s − 0.192·27-s + 1.10·29-s + 0.389·31-s + 0.224·33-s − 1.34·37-s + 0.321·39-s + 1.79·41-s − 1.64·43-s + 0.292·45-s − 0.283·47-s + 0.647·51-s − 0.423·53-s − 0.341·55-s − 0.510·57-s − 1.32·59-s + 1.19·61-s − 0.489·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 - 9.82T + 125T^{2} \)
11 \( 1 + 14.1T + 1.33e3T^{2} \)
13 \( 1 + 26.1T + 2.19e3T^{2} \)
17 \( 1 + 78.5T + 4.91e3T^{2} \)
19 \( 1 - 73.1T + 6.85e3T^{2} \)
23 \( 1 + 96T + 1.21e4T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 - 67.2T + 2.97e4T^{2} \)
37 \( 1 + 301.T + 5.06e4T^{2} \)
41 \( 1 - 472.T + 6.89e4T^{2} \)
43 \( 1 + 463.T + 7.95e4T^{2} \)
47 \( 1 + 91.1T + 1.03e5T^{2} \)
53 \( 1 + 163.T + 1.48e5T^{2} \)
59 \( 1 + 600.T + 2.05e5T^{2} \)
61 \( 1 - 571.T + 2.26e5T^{2} \)
67 \( 1 + 539.T + 3.00e5T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + 442.T + 3.89e5T^{2} \)
79 \( 1 + 45.7T + 4.93e5T^{2} \)
83 \( 1 + 686.T + 5.71e5T^{2} \)
89 \( 1 - 660.T + 7.04e5T^{2} \)
97 \( 1 - 658.T + 9.12e5T^{2} \)
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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

−9.978754787555343134085683454326, −9.164599289499779074575478207739, −8.047569945955533689784127294872, −6.98673762847700125848783934190, −6.13918555541203747343483691575, −5.29034577231318367990490530520, −4.37490867951764857594552708454, −2.78990180525466626535874346812, −1.62445215325896474259516055578, 0, 1.62445215325896474259516055578, 2.78990180525466626535874346812, 4.37490867951764857594552708454, 5.29034577231318367990490530520, 6.13918555541203747343483691575, 6.98673762847700125848783934190, 8.047569945955533689784127294872, 9.164599289499779074575478207739, 9.978754787555343134085683454326

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