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Properties

Label 2-2888-152.13-c0-0-1
Degree 22
Conductor 28882888
Sign 0.158+0.987i0.158 + 0.987i
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  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (0.499 − 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (0.499 − 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((0.158+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((0.158+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: 0.158+0.987i0.158 + 0.987i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(2293,⋅)\chi_{2888} (2293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), 0.158+0.987i)(2,\ 2888,\ (\ :0),\ 0.158 + 0.987i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5709374692.570937469
L(12)L(\frac12) ≈\approx 2.5709374692.570937469
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+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
19 1 1
good3 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
5 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)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+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
17 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
23 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
29 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)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+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
43 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
47 1+(−1.53−1.28i)T+(0.173+0.984i)T2 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2}
53 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
59 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
61 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
67 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
79 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
97 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)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

−9.039595094799104568286263309951, −7.973050191379790988848004046344, −7.48132727229362027685125182894, −6.14714849306925550611210927578, −5.77761721116739746037374203978, −4.34630084337442517755839772711, −4.07017028133501464171808483539, −3.26466141831699413286017230888, −2.29879396054893232067188507500, −1.23975014860829332595279265172, 2.04425037369161743377614303855, 2.54569677714035643723894144212, 3.59642277608225738382707716345, 4.41879634810256802453526232497, 5.34632040136139202675750114620, 6.05470782512826033087256583503, 6.84175561773707240901384118070, 7.71214043812003690708958519178, 8.278669968715083776459683061624, 8.848601459222200494409246312141

Graph of the ZZ-function along the critical line