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Properties

Label 2-1332-148.67-c0-0-0
Degree 22
Conductor 13321332
Sign 0.165+0.986i0.165 + 0.986i
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.85 − 0.326i)5-s + (0.866 + 0.500i)8-s + (−0.939 − 1.62i)10-s + (−1.11 − 1.32i)13-s + (0.173 − 0.984i)16-s + (−0.223 + 0.266i)17-s + (−1.20 + 1.43i)20-s + (2.37 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (1.32 + 0.766i)29-s + (−0.984 + 0.173i)32-s + (0.326 + 0.118i)34-s + (−0.766 + 0.642i)37-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.85 − 0.326i)5-s + (0.866 + 0.500i)8-s + (−0.939 − 1.62i)10-s + (−1.11 − 1.32i)13-s + (0.173 − 0.984i)16-s + (−0.223 + 0.266i)17-s + (−1.20 + 1.43i)20-s + (2.37 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (1.32 + 0.766i)29-s + (−0.984 + 0.173i)32-s + (0.326 + 0.118i)34-s + (−0.766 + 0.642i)37-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((0.165+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=((0.165+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: 0.165+0.986i0.165 + 0.986i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(955,⋅)\chi_{1332} (955, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), 0.165+0.986i)(2,\ 1332,\ (\ :0),\ 0.165 + 0.986i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1202268841.120226884
L(12)L(\frac12) ≈\approx 1.1202268841.120226884
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+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
3 1 1
37 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
good5 1+(−1.85+0.326i)T+(0.939−0.342i)T2 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2}
7 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(1.11+1.32i)T+(−0.173+0.984i)T2 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2}
17 1+(0.223−0.266i)T+(−0.173−0.984i)T2 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2}
19 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1+(−1.32−0.766i)T+(0.5+0.866i)T2 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2}
31 1+T2 1 + T^{2}
41 1+(−1.50+1.26i)T+(0.173−0.984i)T2 1 + (-1.50 + 1.26i)T + (0.173 - 0.984i)T^{2}
43 1+T2 1 + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.300−1.70i)T+(−0.939−0.342i)T2 1 + (0.300 - 1.70i)T + (-0.939 - 0.342i)T^{2}
59 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
61 1+(0.826+0.984i)T+(−0.173+0.984i)T2 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2}
67 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
83 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
89 1+(−0.342−0.0603i)T+(0.939+0.342i)T2 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2}
97 1+(1.11−0.642i)T+(0.5−0.866i)T2 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)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

−9.711495129028565444325372926752, −9.125954577152765826459937964749, −8.340466529275402217937170745385, −7.33682139493869176408948054488, −6.16505334215271097162321452910, −5.27907241930703673834405014677, −4.65446117706914957695090281278, −3.04852952161601770219063159311, −2.36439287260751885014870047146, −1.24404624598771149896843958942, 1.63821338709293107296953715705, 2.62319258197596189405011860772, 4.44488306622822276908806842747, 5.15046387891115202152039021574, 6.09369291918259737644122333065, 6.63382691298596714476923597857, 7.32263898724958685795112335475, 8.504243170977355564834340026565, 9.358207018386588814497372900770, 9.717363340882412989217334763263

Graph of the ZZ-function along the critical line