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Properties

Label 2-2888-152.109-c0-0-1
Degree 22
Conductor 28882888
Sign 0.378+0.925i0.378 + 0.925i
Analytic cond. 1.441291.44129
Root an. cond. 1.200541.20054
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((0.378+0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((0.378+0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: 0.378+0.925i0.378 + 0.925i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(1021,⋅)\chi_{2888} (1021, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), 0.378+0.925i)(2,\ 2888,\ (\ :0),\ 0.378 + 0.925i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5922262482.592226248
L(12)L(\frac12) ≈\approx 2.5922262482.592226248
L(1)L(1) 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+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
19 1 1
good3 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
5 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
7 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}
17 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
23 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
29 1+(0.939+0.342i)T+(0.766+0.642i)T2 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+2T+T2 1 + 2T + T^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
47 1+(1.87+0.684i)T+(0.766+0.642i)T2 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2}
53 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
59 1+(0.939−0.342i)T+(0.766−0.642i)T2 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}
61 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
67 1+(0.939+0.342i)T+(0.766+0.642i)T2 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}
71 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
73 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
79 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
97 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{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

−8.744363192784930897321500821018, −7.82204998226792337518212212441, −7.18615924972471862552597952745, −6.52774437326228030241990723077, −5.60388848093231338754180399671, −5.09450502804641131273600086334, −3.98551967198951257638640478051, −3.10786230711116819668335042260, −1.89787353921309520428918247805, −1.62497127028816034218825278292, 1.59592751060580728261813543724, 3.06206898268024071980378589579, 3.68248748438174691994156191238, 4.39969466412613144149664653094, 5.05149506919231080578216474707, 5.84442100472926051764147726702, 6.74546967741827125935311857998, 7.62659828793603534942575678213, 8.123606209540318977044892358585, 8.995862438182720560129959864959

Graph of the ZZ-function along the critical line