Properties

Label 2-588-1.1-c3-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 1.44·5-s + 9·9-s + 46.1·11-s − 32.2·13-s − 4.33·15-s + 77.7·17-s + 12.6·19-s − 100.·23-s − 122.·25-s + 27·27-s + 213.·29-s + 42.0·31-s + 138.·33-s + 310.·37-s − 96.6·39-s + 44.0·41-s + 381.·43-s − 13.0·45-s − 358.·47-s + 233.·51-s + 184.·53-s − 66.7·55-s + 37.9·57-s + 454.·59-s + 11.8·61-s + 46.6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.129·5-s + 0.333·9-s + 1.26·11-s − 0.687·13-s − 0.0746·15-s + 1.10·17-s + 0.152·19-s − 0.914·23-s − 0.983·25-s + 0.192·27-s + 1.36·29-s + 0.243·31-s + 0.729·33-s + 1.37·37-s − 0.397·39-s + 0.167·41-s + 1.35·43-s − 0.0431·45-s − 1.11·47-s + 0.640·51-s + 0.479·53-s − 0.163·55-s + 0.0882·57-s + 1.00·59-s + 0.0248·61-s + 0.0889·65-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: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.626865615\)
\(L(\frac12)\) \(\approx\) \(2.626865615\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.44T + 125T^{2} \)
11 \( 1 - 46.1T + 1.33e3T^{2} \)
13 \( 1 + 32.2T + 2.19e3T^{2} \)
17 \( 1 - 77.7T + 4.91e3T^{2} \)
19 \( 1 - 12.6T + 6.85e3T^{2} \)
23 \( 1 + 100.T + 1.21e4T^{2} \)
29 \( 1 - 213.T + 2.43e4T^{2} \)
31 \( 1 - 42.0T + 2.97e4T^{2} \)
37 \( 1 - 310.T + 5.06e4T^{2} \)
41 \( 1 - 44.0T + 6.89e4T^{2} \)
43 \( 1 - 381.T + 7.95e4T^{2} \)
47 \( 1 + 358.T + 1.03e5T^{2} \)
53 \( 1 - 184.T + 1.48e5T^{2} \)
59 \( 1 - 454.T + 2.05e5T^{2} \)
61 \( 1 - 11.8T + 2.26e5T^{2} \)
67 \( 1 - 590.T + 3.00e5T^{2} \)
71 \( 1 - 494.T + 3.57e5T^{2} \)
73 \( 1 - 975.T + 3.89e5T^{2} \)
79 \( 1 - 299.T + 4.93e5T^{2} \)
83 \( 1 + 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + 695.T + 7.04e5T^{2} \)
97 \( 1 - 481.T + 9.12e5T^{2} \)
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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

−9.860396495451413029185168266246, −9.672644203229253540261957427796, −8.435716917614640016286754920283, −7.75786036666844855193264594627, −6.76846647949485184499345955135, −5.78069939949090106550768378441, −4.45403947814252433894151747473, −3.60131905047754698392058913606, −2.36576510199126841404761466631, −0.994625576613004797376751722141, 0.994625576613004797376751722141, 2.36576510199126841404761466631, 3.60131905047754698392058913606, 4.45403947814252433894151747473, 5.78069939949090106550768378441, 6.76846647949485184499345955135, 7.75786036666844855193264594627, 8.435716917614640016286754920283, 9.672644203229253540261957427796, 9.860396495451413029185168266246

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