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Properties

Label 2-1512-1512.1021-c0-0-1
Degree 22
Conductor 15121512
Sign 0.448−0.893i0.448 - 0.893i
Analytic cond. 0.7545860.754586
Root an. cond. 0.8686690.868669
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 + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−1.76 + 0.642i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.939 + 1.62i)10-s + (0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−1.43 − 1.20i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)18-s + (0.173 + 0.300i)19-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−1.76 + 0.642i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.939 + 1.62i)10-s + (0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−1.43 − 1.20i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)18-s + (0.173 + 0.300i)19-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((0.448−0.893i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s) L(s)=((0.448−0.893i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 0.448−0.893i0.448 - 0.893i
Analytic conductor: 0.7545860.754586
Root analytic conductor: 0.8686690.868669
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1512(1021,⋅)\chi_{1512} (1021, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :0), 0.448−0.893i)(2,\ 1512,\ (\ :0),\ 0.448 - 0.893i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3495578451.349557845
L(12)L(\frac12) ≈\approx 1.3495578451.349557845
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
3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
good5 1+(1.76−0.642i)T+(0.766−0.642i)T2 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2}
11 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
13 1+(−0.766−0.642i)T+(0.173+0.984i)T2 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.173−0.300i)T+(−0.5+0.866i)T2 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
53 1−T2 1 - T^{2}
59 1+(−1.87+0.684i)T+(0.766−0.642i)T2 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2}
61 1+(0.266+1.50i)T+(−0.939+0.342i)T2 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2}
67 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
71 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.266+0.223i)T+(0.173−0.984i)T2 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2}
83 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−9.886444321101022172110579428437, −9.114952655861471386966700665097, −8.295381787836537238964274245200, −7.52678742585252470890586612506, −6.40596660251561964991236748421, −5.37672425588564750765151350971, −4.51622457574136755048623136667, −3.59287874635185335457275662004, −3.32791386716660817160323228668, −2.05265841945606898454214609829, 0.826082195581700843991105819901, 2.79358374711878727284996130663, 3.81824972038186660433067743128, 4.22235370145455635383042878382, 5.32630609007917896063392875805, 6.59450438517177953073031604181, 7.13738748432169065925737663307, 7.957126945048532260898093504836, 8.267377806203882910330444074930, 8.930169819553289886574223132285

Graph of the ZZ-function along the critical line