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Properties

Label 2-588-21.20-c3-0-34
Degree 22
Conductor 588588
Sign −0.571−0.820i-0.571 - 0.820i
Analytic cond. 34.693134.6931
Root an. cond. 5.890085.89008
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.03 − 1.28i)3-s − 19.3·5-s + (23.7 + 12.9i)9-s − 41.9i·11-s − 77.1i·13-s + (97.5 + 24.8i)15-s − 5.96·17-s − 68.1i·19-s + 35.5i·23-s + 250.·25-s + (−102. − 95.4i)27-s − 228. i·29-s − 72.0i·31-s + (−53.7 + 211. i)33-s + 69.3·37-s + ⋯
L(s)  = 1  + (−0.969 − 0.246i)3-s − 1.73·5-s + (0.878 + 0.478i)9-s − 1.14i·11-s − 1.64i·13-s + (1.67 + 0.427i)15-s − 0.0851·17-s − 0.823i·19-s + 0.322i·23-s + 2.00·25-s + (−0.733 − 0.680i)27-s − 1.46i·29-s − 0.417i·31-s + (−0.283 + 1.11i)33-s + 0.308·37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((−0.571−0.820i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+3/2) L(s)=((−0.571−0.820i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −0.571−0.820i-0.571 - 0.820i
Analytic conductor: 34.693134.6931
Root analytic conductor: 5.890085.89008
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ588(293,⋅)\chi_{588} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :3/2), −0.571−0.820i)(2,\ 588,\ (\ :3/2),\ -0.571 - 0.820i)

Particular Values

L(2)L(2) ≈\approx 0.21421429430.2142142943
L(12)L(\frac12) ≈\approx 0.21421429430.2142142943
L(52)L(\frac{5}{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
3 1+(5.03+1.28i)T 1 + (5.03 + 1.28i)T
7 1 1
good5 1+19.3T+125T2 1 + 19.3T + 125T^{2}
11 1+41.9iT−1.33e3T2 1 + 41.9iT - 1.33e3T^{2}
13 1+77.1iT−2.19e3T2 1 + 77.1iT - 2.19e3T^{2}
17 1+5.96T+4.91e3T2 1 + 5.96T + 4.91e3T^{2}
19 1+68.1iT−6.85e3T2 1 + 68.1iT - 6.85e3T^{2}
23 1−35.5iT−1.21e4T2 1 - 35.5iT - 1.21e4T^{2}
29 1+228.iT−2.43e4T2 1 + 228. iT - 2.43e4T^{2}
31 1+72.0iT−2.97e4T2 1 + 72.0iT - 2.97e4T^{2}
37 1−69.3T+5.06e4T2 1 - 69.3T + 5.06e4T^{2}
41 1+132.T+6.89e4T2 1 + 132.T + 6.89e4T^{2}
43 1+366.T+7.95e4T2 1 + 366.T + 7.95e4T^{2}
47 1+181.T+1.03e5T2 1 + 181.T + 1.03e5T^{2}
53 1+17.6iT−1.48e5T2 1 + 17.6iT - 1.48e5T^{2}
59 1−494.T+2.05e5T2 1 - 494.T + 2.05e5T^{2}
61 1−575.iT−2.26e5T2 1 - 575. iT - 2.26e5T^{2}
67 1+404.T+3.00e5T2 1 + 404.T + 3.00e5T^{2}
71 1+489.iT−3.57e5T2 1 + 489. iT - 3.57e5T^{2}
73 1−388.iT−3.89e5T2 1 - 388. iT - 3.89e5T^{2}
79 1−253.T+4.93e5T2 1 - 253.T + 4.93e5T^{2}
83 1−356.T+5.71e5T2 1 - 356.T + 5.71e5T^{2}
89 1+1.42e3T+7.04e5T2 1 + 1.42e3T + 7.04e5T^{2}
97 1+694.iT−9.12e5T2 1 + 694. iT - 9.12e5T^{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

−9.994211904922761320733977532370, −8.462050842371205266875372181571, −7.923845455399238293913514421830, −7.11124541081034382708186468005, −6.01328956649969192153837070827, −5.07621451728668222722268913408, −4.01955592451829000900086176342, −3.00456969659347311028396468760, −0.76627010068147160927620732579, −0.11453443179594399573839156868, 1.56062336692726040960130452546, 3.59681620869548234142096524272, 4.36996449417082036924796199386, 5.03887137887832541291397941681, 6.69074696986221005520547494652, 7.04446458809750726323383624337, 8.113832504746327648101384297419, 9.161566480769155410173346030567, 10.15672568949937118726117703207, 11.04171825872909376862291324917

Graph of the ZZ-function along the critical line