Properties

Label 2-588-1.1-c3-0-17
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 + 16.6·5-s + 9·9-s − 71.7·11-s + 65.3·13-s − 49.9·15-s − 90.4·17-s − 163.·19-s + 79.2·23-s + 152.·25-s − 27·27-s − 43.2·29-s − 135.·31-s + 215.·33-s + 270.·37-s − 196.·39-s − 152.·41-s − 177.·43-s + 149.·45-s + 45.6·47-s + 271.·51-s − 158.·53-s − 1.19e3·55-s + 491.·57-s + 391.·59-s − 551.·61-s + 1.08e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.48·5-s + 0.333·9-s − 1.96·11-s + 1.39·13-s − 0.860·15-s − 1.29·17-s − 1.97·19-s + 0.718·23-s + 1.21·25-s − 0.192·27-s − 0.277·29-s − 0.785·31-s + 1.13·33-s + 1.20·37-s − 0.805·39-s − 0.579·41-s − 0.630·43-s + 0.496·45-s + 0.141·47-s + 0.744·51-s − 0.410·53-s − 2.93·55-s + 1.14·57-s + 0.864·59-s − 1.15·61-s + 2.07·65-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 \( 1 - 16.6T + 125T^{2} \)
11 \( 1 + 71.7T + 1.33e3T^{2} \)
13 \( 1 - 65.3T + 2.19e3T^{2} \)
17 \( 1 + 90.4T + 4.91e3T^{2} \)
19 \( 1 + 163.T + 6.85e3T^{2} \)
23 \( 1 - 79.2T + 1.21e4T^{2} \)
29 \( 1 + 43.2T + 2.43e4T^{2} \)
31 \( 1 + 135.T + 2.97e4T^{2} \)
37 \( 1 - 270.T + 5.06e4T^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + 177.T + 7.95e4T^{2} \)
47 \( 1 - 45.6T + 1.03e5T^{2} \)
53 \( 1 + 158.T + 1.48e5T^{2} \)
59 \( 1 - 391.T + 2.05e5T^{2} \)
61 \( 1 + 551.T + 2.26e5T^{2} \)
67 \( 1 - 458.T + 3.00e5T^{2} \)
71 \( 1 + 486.T + 3.57e5T^{2} \)
73 \( 1 + 574.T + 3.89e5T^{2} \)
79 \( 1 + 668.T + 4.93e5T^{2} \)
83 \( 1 + 76.2T + 5.71e5T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 242.T + 9.12e5T^{2} \)
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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

−10.07412223371146558261917217110, −8.989341865408104736811941100916, −8.246749278458087816664197861475, −6.85173273663045907348056130972, −6.07832588411505786965066831512, −5.41916105093184054402285194947, −4.38687920095016036528331189445, −2.66521454988000789994205161543, −1.72924220673063275512560032672, 0, 1.72924220673063275512560032672, 2.66521454988000789994205161543, 4.38687920095016036528331189445, 5.41916105093184054402285194947, 6.07832588411505786965066831512, 6.85173273663045907348056130972, 8.246749278458087816664197861475, 8.989341865408104736811941100916, 10.07412223371146558261917217110

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