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Properties

Label 2-5780-17.16-c1-0-39
Degree 22
Conductor 57805780
Sign 0.410+0.911i0.410 + 0.911i
Analytic cond. 46.153546.1535
Root an. cond. 6.793636.79363
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  − 2.30i·3-s + i·5-s + 1.53i·7-s − 2.30·9-s + 2.44i·11-s − 6.18·13-s + 2.30·15-s + 0.345·19-s + 3.52·21-s − 9.04i·23-s − 25-s − 1.60i·27-s + 5.05i·29-s + 2.71i·31-s + 5.63·33-s + ⋯
L(s)  = 1  − 1.32i·3-s + 0.447i·5-s + 0.579i·7-s − 0.767·9-s + 0.738i·11-s − 1.71·13-s + 0.594·15-s + 0.0793·19-s + 0.769·21-s − 1.88i·23-s − 0.200·25-s − 0.308i·27-s + 0.938i·29-s + 0.486i·31-s + 0.981·33-s + ⋯

Functional equation

Λ(s)=(5780s/2 ΓC(s) L(s)=((0.410+0.911i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5780s/2 ΓC(s+1/2) L(s)=((0.410+0.911i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57805780    =    22⋅5⋅1722^{2} \cdot 5 \cdot 17^{2}
Sign: 0.410+0.911i0.410 + 0.911i
Analytic conductor: 46.153546.1535
Root analytic conductor: 6.793636.79363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5780(5201,⋅)\chi_{5780} (5201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5780, ( :1/2), 0.410+0.911i)(2,\ 5780,\ (\ :1/2),\ 0.410 + 0.911i)

Particular Values

L(1)L(1) ≈\approx 1.5139032081.513903208
L(12)L(\frac12) ≈\approx 1.5139032081.513903208
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
5 1−iT 1 - iT
17 1 1
good3 1+2.30iT−3T2 1 + 2.30iT - 3T^{2}
7 1−1.53iT−7T2 1 - 1.53iT - 7T^{2}
11 1−2.44iT−11T2 1 - 2.44iT - 11T^{2}
13 1+6.18T+13T2 1 + 6.18T + 13T^{2}
19 1−0.345T+19T2 1 - 0.345T + 19T^{2}
23 1+9.04iT−23T2 1 + 9.04iT - 23T^{2}
29 1−5.05iT−29T2 1 - 5.05iT - 29T^{2}
31 1−2.71iT−31T2 1 - 2.71iT - 31T^{2}
37 1−4.08iT−37T2 1 - 4.08iT - 37T^{2}
41 1−1.14iT−41T2 1 - 1.14iT - 41T^{2}
43 1+0.730T+43T2 1 + 0.730T + 43T^{2}
47 1−0.594T+47T2 1 - 0.594T + 47T^{2}
53 1−9.59T+53T2 1 - 9.59T + 53T^{2}
59 1+6.74T+59T2 1 + 6.74T + 59T^{2}
61 1+10.5iT−61T2 1 + 10.5iT - 61T^{2}
67 1−12.3T+67T2 1 - 12.3T + 67T^{2}
71 1+1.08iT−71T2 1 + 1.08iT - 71T^{2}
73 1+9.34iT−73T2 1 + 9.34iT - 73T^{2}
79 1+11.0iT−79T2 1 + 11.0iT - 79T^{2}
83 1−12.2T+83T2 1 - 12.2T + 83T^{2}
89 1+10.8T+89T2 1 + 10.8T + 89T^{2}
97 1+6.97iT−97T2 1 + 6.97iT - 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

−7.77975553480604102207728928917, −7.19478664656751994390089730872, −6.74065131060740363997320951865, −6.11868287843021566638521971557, −5.06833929298112410562496119840, −4.55193722397575006284503931455, −3.16374752074263549960604820073, −2.35000204418516994607181415005, −1.94174754753309377127823620623, −0.54872758305278421388028890360, 0.73182691918898281799742053767, 2.17352207488376703431927761323, 3.20568799675767873080319311155, 3.99593778532281431890572818747, 4.45462429009808348939356083208, 5.42893500431320514140006235586, 5.62883532779518287794389279656, 7.01025622796444784344242201226, 7.53316308928049655043086178483, 8.309193080517299212689390768079

Graph of the ZZ-function along the critical line