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Properties

Label 2-588-12.11-c1-0-39
Degree 22
Conductor 588588
Sign 0.942+0.334i0.942 + 0.334i
Analytic cond. 4.695204.69520
Root an. cond. 2.166842.16684
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(588s/2 ΓC(s) L(s)=((0.942+0.334i) Λ‾(2−s)\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}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=((0.942+0.334i) Λ‾(1−s)\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: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.942+0.334i0.942 + 0.334i
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ588(491,⋅)\chi_{588} (491, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :1/2), 0.942+0.334i)(2,\ 588,\ (\ :1/2),\ 0.942 + 0.334i)

Particular Values

L(1)L(1) ≈\approx 2.29247−0.394379i2.29247 - 0.394379i
L(12)L(\frac12) ≈\approx 2.29247−0.394379i2.29247 - 0.394379i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−1.38+0.300i)T 1 + (-1.38 + 0.300i)T
3 1+(1.24+1.20i)T 1 + (1.24 + 1.20i)T
7 1 1
good5 1−2.72iT−5T2 1 - 2.72iT - 5T^{2}
11 1−2.04T+11T2 1 - 2.04T + 11T^{2}
13 1−4.44T+13T2 1 - 4.44T + 13T^{2}
17 1−0.660iT−17T2 1 - 0.660iT - 17T^{2}
19 1+2.93iT−19T2 1 + 2.93iT - 19T^{2}
23 1−1.04T+23T2 1 - 1.04T + 23T^{2}
29 1−2.06iT−29T2 1 - 2.06iT - 29T^{2}
31 1−3.10iT−31T2 1 - 3.10iT - 31T^{2}
37 1+9.52T+37T2 1 + 9.52T + 37T^{2}
41 1+7.85iT−41T2 1 + 7.85iT - 41T^{2}
43 1+0.530iT−43T2 1 + 0.530iT - 43T^{2}
47 1−1.04T+47T2 1 - 1.04T + 47T^{2}
53 1−10.5iT−53T2 1 - 10.5iT - 53T^{2}
59 1+3.48T+59T2 1 + 3.48T + 59T^{2}
61 1−0.198T+61T2 1 - 0.198T + 61T^{2}
67 1−4.76iT−67T2 1 - 4.76iT - 67T^{2}
71 1+5.52T+71T2 1 + 5.52T + 71T^{2}
73 1−3.03T+73T2 1 - 3.03T + 73T^{2}
79 1+10.9iT−79T2 1 + 10.9iT - 79T^{2}
83 1+9.15T+83T2 1 + 9.15T + 83T^{2}
89 1+0.541iT−89T2 1 + 0.541iT - 89T^{2}
97 1−6.91T+97T2 1 - 6.91T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line