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Properties

Label 2-3042-13.12-c1-0-51
Degree 22
Conductor 30423042
Sign −0.960+0.277i-0.960 + 0.277i
Analytic cond. 24.290424.2904
Root an. cond. 4.928534.92853
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  − i·2-s − 4-s − 0.267i·5-s − 0.732i·7-s + i·8-s − 0.267·10-s + 4.73i·11-s − 0.732·14-s + 16-s − 2.26·17-s + 1.26i·19-s + 0.267i·20-s + 4.73·22-s − 6.19·23-s + 4.92·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.119i·5-s − 0.276i·7-s + 0.353i·8-s − 0.0847·10-s + 1.42i·11-s − 0.195·14-s + 0.250·16-s − 0.550·17-s + 0.290i·19-s + 0.0599i·20-s + 1.00·22-s − 1.29·23-s + 0.985·25-s + ⋯

Functional equation

Λ(s)=(3042s/2 ΓC(s) L(s)=((−0.960+0.277i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3042s/2 ΓC(s+1/2) L(s)=((−0.960+0.277i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30423042    =    2⋅32⋅1322 \cdot 3^{2} \cdot 13^{2}
Sign: −0.960+0.277i-0.960 + 0.277i
Analytic conductor: 24.290424.2904
Root analytic conductor: 4.928534.92853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3042(1351,⋅)\chi_{3042} (1351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3042, ( :1/2), −0.960+0.277i)(2,\ 3042,\ (\ :1/2),\ -0.960 + 0.277i)

Particular Values

L(1)L(1) ≈\approx 0.80261728920.8026172892
L(12)L(\frac12) ≈\approx 0.80261728920.8026172892
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+iT 1 + iT
3 1 1
13 1 1
good5 1+0.267iT−5T2 1 + 0.267iT - 5T^{2}
7 1+0.732iT−7T2 1 + 0.732iT - 7T^{2}
11 1−4.73iT−11T2 1 - 4.73iT - 11T^{2}
17 1+2.26T+17T2 1 + 2.26T + 17T^{2}
19 1−1.26iT−19T2 1 - 1.26iT - 19T^{2}
23 1+6.19T+23T2 1 + 6.19T + 23T^{2}
29 1+2.46T+29T2 1 + 2.46T + 29T^{2}
31 1+5.46iT−31T2 1 + 5.46iT - 31T^{2}
37 1+10.4iT−37T2 1 + 10.4iT - 37T^{2}
41 1+11.3iT−41T2 1 + 11.3iT - 41T^{2}
43 1+7.66T+43T2 1 + 7.66T + 43T^{2}
47 1+8.19iT−47T2 1 + 8.19iT - 47T^{2}
53 1+0.464T+53T2 1 + 0.464T + 53T^{2}
59 1−8iT−59T2 1 - 8iT - 59T^{2}
61 1−1.19T+61T2 1 - 1.19T + 61T^{2}
67 1+11.1iT−67T2 1 + 11.1iT - 67T^{2}
71 1+1.26iT−71T2 1 + 1.26iT - 71T^{2}
73 1−9.73iT−73T2 1 - 9.73iT - 73T^{2}
79 1+9.46T+79T2 1 + 9.46T + 79T^{2}
83 1+10.1iT−83T2 1 + 10.1iT - 83T^{2}
89 1+2.53iT−89T2 1 + 2.53iT - 89T^{2}
97 1+6iT−97T2 1 + 6iT - 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

−8.594481150303802868182784752399, −7.53862578667208030779271220572, −7.09094581939577671757210439272, −5.97920201980613615158359463612, −5.13249255116510348281264422059, −4.24551169775977822562442416109, −3.75127391137503807564301600127, −2.36840272403895561543210616928, −1.79636515511229659383477890065, −0.26085182800569717763175867203, 1.22908884002359873655489298257, 2.73994971371760346649848996072, 3.51452518265488197486432134784, 4.58682070005904547483214069612, 5.30487877025916276875374820309, 6.26083451395660273360724818217, 6.54685230359560823149645254826, 7.62209353850871041828246016213, 8.418488972189044625916330325170, 8.716727167578236702309870022480

Graph of the ZZ-function along the critical line