Properties

Label 2-588-1.1-c3-0-2
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 − 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: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.389376864\)
\(L(\frac12)\) \(\approx\) \(1.389376864\)
\(L(\frac{5}{2})\) 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

−10.15572147027195724711433693865, −9.535914548885003556622725089612, −8.125143779138036233853995573517, −7.70227496230619663992881668772, −7.26364555242624219554171899662, −5.38413425351335828253403332730, −4.70935503286620181948025040039, −3.32462742399688347107116939878, −2.71743087310872876435127422539, −0.66849252999246936064228612097, 0.66849252999246936064228612097, 2.71743087310872876435127422539, 3.32462742399688347107116939878, 4.70935503286620181948025040039, 5.38413425351335828253403332730, 7.26364555242624219554171899662, 7.70227496230619663992881668772, 8.125143779138036233853995573517, 9.535914548885003556622725089612, 10.15572147027195724711433693865

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