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Properties

Label 2-588-12.11-c1-0-39
Degree $2$
Conductor $588$
Sign $0.942 + 0.334i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 0.300i)2-s + (−1.24 − 1.20i)3-s + (1.81 − 0.830i)4-s + 2.72i·5-s + (−2.08 − 1.29i)6-s + (2.26 − 1.69i)8-s + (0.0992 + 2.99i)9-s + (0.819 + 3.76i)10-s + 2.04·11-s + (−3.26 − 1.15i)12-s + 4.44·13-s + (3.28 − 3.39i)15-s + (2.62 − 3.02i)16-s + 0.660i·17-s + (1.03 + 4.11i)18-s − 2.93i·19-s + ⋯
L(s)  = 1  + (0.977 − 0.212i)2-s + (−0.718 − 0.695i)3-s + (0.909 − 0.415i)4-s + 1.21i·5-s + (−0.850 − 0.526i)6-s + (0.800 − 0.598i)8-s + (0.0330 + 0.999i)9-s + (0.259 + 1.19i)10-s + 0.615·11-s + (−0.942 − 0.334i)12-s + 1.23·13-s + (0.848 − 0.876i)15-s + (0.655 − 0.755i)16-s + 0.160i·17-s + (0.244 + 0.969i)18-s − 0.674i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29247 - 0.394379i\)
\(L(\frac12)\) \(\approx\) \(2.29247 - 0.394379i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.300i)T \)
3 \( 1 + (1.24 + 1.20i)T \)
7 \( 1 \)
good5 \( 1 - 2.72iT - 5T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 - 0.660iT - 17T^{2} \)
19 \( 1 + 2.93iT - 19T^{2} \)
23 \( 1 - 1.04T + 23T^{2} \)
29 \( 1 - 2.06iT - 29T^{2} \)
31 \( 1 - 3.10iT - 31T^{2} \)
37 \( 1 + 9.52T + 37T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 + 0.530iT - 43T^{2} \)
47 \( 1 - 1.04T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + 3.48T + 59T^{2} \)
61 \( 1 - 0.198T + 61T^{2} \)
67 \( 1 - 4.76iT - 67T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 - 3.03T + 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + 0.541iT - 89T^{2} \)
97 \( 1 - 6.91T + 97T^{2} \)
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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

−10.70695196942099242655402735994, −10.53464367542502790131139492012, −8.854625766111534348441352438642, −7.43828925154376613142843628501, −6.77632426792992936385971977882, −6.20358635903159970501427807591, −5.24232416504960752241496291706, −3.91175305111266218202578592569, −2.84200850840767786529063229693, −1.49699081777781396187069506090, 1.36495102300101663330535592059, 3.49637228791575362722761031792, 4.27350532883016236670092201058, 5.14161078791946538862875618621, 5.92376877634595825367659463108, 6.73161787230283413183885341422, 8.161321091727392012355702208547, 8.945649129114342041883205465811, 9.990139160092874133372879410502, 11.04034530852412427930311718002

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