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Properties

Label 2-3042-13.12-c1-0-53
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 − 1.73i·5-s + 4.73i·7-s + i·8-s − 1.73·10-s − 4.73i·11-s + 4.73·14-s + 16-s − 5.19·17-s − 1.26i·19-s + 1.73i·20-s − 4.73·22-s + 2.19·23-s + 2.00·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.774i·5-s + 1.78i·7-s + 0.353i·8-s − 0.547·10-s − 1.42i·11-s + 1.26·14-s + 0.250·16-s − 1.26·17-s − 0.290i·19-s + 0.387i·20-s − 1.00·22-s + 0.457·23-s + 0.400·25-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 0.92616066180.9261606618
L(12)L(\frac12) ≈\approx 0.92616066180.9261606618
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+1.73iT−5T2 1 + 1.73iT - 5T^{2}
7 1−4.73iT−7T2 1 - 4.73iT - 7T^{2}
11 1+4.73iT−11T2 1 + 4.73iT - 11T^{2}
17 1+5.19T+17T2 1 + 5.19T + 17T^{2}
19 1+1.26iT−19T2 1 + 1.26iT - 19T^{2}
23 1−2.19T+23T2 1 - 2.19T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1+2.53iT−31T2 1 + 2.53iT - 31T^{2}
37 1−3iT−37T2 1 - 3iT - 37T^{2}
41 1+0.464iT−41T2 1 + 0.464iT - 41T^{2}
43 1+6.19T+43T2 1 + 6.19T + 43T^{2}
47 1−1.26iT−47T2 1 - 1.26iT - 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1+13.8iT−59T2 1 + 13.8iT - 59T^{2}
61 1−4.80T+61T2 1 - 4.80T + 61T^{2}
67 1+10.7iT−67T2 1 + 10.7iT - 67T^{2}
71 1−8.19iT−71T2 1 - 8.19iT - 71T^{2}
73 1+12.1iT−73T2 1 + 12.1iT - 73T^{2}
79 1+12.3T+79T2 1 + 12.3T + 79T^{2}
83 1+11.6iT−83T2 1 + 11.6iT - 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

−8.580080998641710208433201245408, −8.151216046749202505879480909086, −6.63320076716790490822361960942, −5.94968681617951073683584634279, −5.15967365969020026069904649934, −4.60839319173876106184715581880, −3.31088023549051013013276678955, −2.64823604294301214148879193302, −1.66520142302483777679319300576, −0.30300188091268477826440440497, 1.26387557611753727194401765180, 2.61812365475114739270998892494, 3.82078439526043869015774741772, 4.38386767823403846317582874454, 5.10717238815327686419250091804, 6.43710574658513813850490025565, 7.00841506986143513427241358296, 7.18958337044036658506612967423, 8.081654031349349833089996492157, 8.990815706107538061250050632790

Graph of the ZZ-function along the critical line