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Properties

Label 2-588-21.17-c3-0-13
Degree $2$
Conductor $588$
Sign $-0.262 - 0.964i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.15 + 0.637i)3-s + (6.99 + 12.1i)5-s + (26.1 + 6.57i)9-s + (−27.5 − 15.8i)11-s + 31.1i·13-s + (28.3 + 66.9i)15-s + (−61.5 + 106. i)17-s + (−62.7 + 36.2i)19-s + (66.1 − 38.1i)23-s + (−35.3 + 61.1i)25-s + (130. + 50.6i)27-s − 14.1i·29-s + (267. + 154. i)31-s + (−131. − 99.5i)33-s + (58.3 + 101. i)37-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)3-s + (0.625 + 1.08i)5-s + (0.969 + 0.243i)9-s + (−0.754 − 0.435i)11-s + 0.663i·13-s + (0.487 + 1.15i)15-s + (−0.877 + 1.52i)17-s + (−0.757 + 0.437i)19-s + (0.599 − 0.346i)23-s + (−0.282 + 0.489i)25-s + (0.932 + 0.360i)27-s − 0.0903i·29-s + (1.55 + 0.896i)31-s + (−0.695 − 0.525i)33-s + (0.259 + 0.449i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.729637870\)
\(L(\frac12)\) \(\approx\) \(2.729637870\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.15 - 0.637i)T \)
7 \( 1 \)
good5 \( 1 + (-6.99 - 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (27.5 + 15.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 31.1iT - 2.19e3T^{2} \)
17 \( 1 + (61.5 - 106. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (62.7 - 36.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-66.1 + 38.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 14.1iT - 2.43e4T^{2} \)
31 \( 1 + (-267. - 154. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-58.3 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 491.T + 6.89e4T^{2} \)
43 \( 1 - 13.6T + 7.95e4T^{2} \)
47 \( 1 + (43.4 + 75.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (388. + 224. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-128. + 222. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-238. + 137. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (515. - 893. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 931. iT - 3.57e5T^{2} \)
73 \( 1 + (-887. - 512. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-462. - 800. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 991.T + 5.71e5T^{2} \)
89 \( 1 + (-125. - 217. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−10.43831820419857533928578405314, −9.829259183335622291559805545040, −8.595541224862188348148112011287, −8.176920954103732386258390713881, −6.76713690554728383783692714419, −6.38299300298538334589625372826, −4.82859764472790034929413183777, −3.65910487449025363588766366983, −2.66335736721880183613599432843, −1.79507854497257640946916916535, 0.66454872507473971648637797458, 2.05501998429056270340613264365, 2.96596340177938402110204191496, 4.55570155861030961094273729719, 5.11164065378283560571203954646, 6.52458582976584452519192477788, 7.55182749246380944973997955955, 8.382475678250548223623609207066, 9.173622368779553223527634262476, 9.727657350501824677939327197037

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