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Properties

Label 2-2888-152.123-c0-0-4
Degree 22
Conductor 28882888
Sign −0.320+0.947i-0.320 + 0.947i
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  + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 0.879·18-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 0.879·18-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((−0.320+0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((−0.320+0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: −0.320+0.947i-0.320 + 0.947i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(2555,⋅)\chi_{2888} (2555, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), −0.320+0.947i)(2,\ 2888,\ (\ :0),\ -0.320 + 0.947i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.75489404630.7548940463
L(12)L(\frac12) ≈\approx 0.75489404630.7548940463
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+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
19 1 1
good3 1+(0.266−0.223i)T+(0.173−0.984i)T2 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2}
5 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
17 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
23 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
29 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−1.43+1.20i)T+(0.173−0.984i)T2 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2}
43 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
47 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
53 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
59 1+(0.266+1.50i)T+(−0.939+0.342i)T2 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2}
61 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
67 1+(−0.326+1.85i)T+(−0.939−0.342i)T2 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(−1.17+0.984i)T+(0.173−0.984i)T2 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2}
79 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
83 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
89 1+(−0.766−0.642i)T+(0.173+0.984i)T2 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}
97 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)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

−8.968237667472593964308513722936, −8.002205171608649250853049570174, −7.61637237882617392739158044035, −6.28694151884376407231523547033, −5.21544600684911770182465188639, −4.99480829377312303487336542660, −3.77082388704072490170544603374, −2.93725894377190229711552888042, −2.15111313930195913063104150678, −0.60647879965516300751839247632, 1.17052212995568422422147514703, 2.61097146011034204308030634218, 3.98539576257835080118937488202, 4.59695926190580993289383868058, 5.55828918769739206273958758874, 6.22540698324612100592540337893, 6.91218848227386259220066398607, 7.57319669663218234230541428815, 8.296315663433176038364362448114, 9.085700874454810343596894067043

Graph of the ZZ-function along the critical line