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Properties

Label 2-456-456.371-c0-0-1
Degree 22
Conductor 456456
Sign 0.877+0.479i0.877 + 0.479i
Analytic cond. 0.2275730.227573
Root an. cond. 0.4770460.477046
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 + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (1.11 − 0.642i)11-s + (−0.939 − 0.342i)12-s + (0.766 − 0.642i)16-s + (−1.70 + 0.300i)17-s + (0.939 − 0.342i)18-s + (0.939 − 0.342i)19-s + (−0.826 − 0.984i)22-s + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (1.11 − 0.642i)11-s + (−0.939 − 0.342i)12-s + (0.766 − 0.642i)16-s + (−1.70 + 0.300i)17-s + (0.939 − 0.342i)18-s + (0.939 − 0.342i)19-s + (−0.826 − 0.984i)22-s + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)25-s + ⋯

Functional equation

Λ(s)=(456s/2 ΓC(s) L(s)=((0.877+0.479i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(456s/2 ΓC(s) L(s)=((0.877+0.479i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 456456    =    23⋅3⋅192^{3} \cdot 3 \cdot 19
Sign: 0.877+0.479i0.877 + 0.479i
Analytic conductor: 0.2275730.227573
Root analytic conductor: 0.4770460.477046
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ456(371,⋅)\chi_{456} (371, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 456, ( :0), 0.877+0.479i)(2,\ 456,\ (\ :0),\ 0.877 + 0.479i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.93149784120.9314978412
L(12)L(\frac12) ≈\approx 0.93149784120.9314978412
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
3 1+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
19 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
good5 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+(−1.11+0.642i)T+(0.5−0.866i)T2 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2}
13 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
17 1+(1.70−0.300i)T+(0.939−0.342i)T2 1 + (1.70 - 0.300i)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+(−0.673−0.118i)T+(0.939+0.342i)T2 1 + (-0.673 - 0.118i)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+(−0.592−0.342i)T+(0.5+0.866i)T2 1 + (-0.592 - 0.342i)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+(−0.673+0.118i)T+(0.939−0.342i)T2 1 + (-0.673 + 0.118i)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

−11.27688028922704793334091761906, −10.23058347715233385500414472807, −9.475539906193342698303658864429, −8.750624527835112901274139890973, −8.089665006437769031637283727172, −6.63597500865994226887285714836, −5.04932523465199771566247850214, −4.07911500863910084336060449241, −3.21818330032895592244622420676, −1.90859637899668004557641393485, 1.70915252033712958453505294437, 3.57717016684326603828098256185, 4.65477982764100097920471162564, 6.08843723779040191813334854047, 6.93599250019069259339813607499, 7.52610850197381885973726946623, 8.670968385659675939929295035158, 9.213843311972728815636261830780, 10.01714259816695836277694633853, 11.52550825316153266248530674854

Graph of the ZZ-function along the critical line