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Properties

Label 2-1800-1800.931-c0-0-0
Degree 22
Conductor 18001800
Sign −0.851+0.523i-0.851 + 0.523i
Analytic cond. 0.8983170.898317
Root an. cond. 0.9477950.947795
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  + (−0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 + 0.994i)4-s + (−0.994 + 0.104i)5-s + (−0.994 − 0.104i)6-s + (−0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.413 + 0.459i)11-s + (0.669 + 0.743i)12-s + (1.20 − 1.08i)13-s + (0.978 + 0.207i)14-s + (−0.743 + 0.669i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 + 0.994i)4-s + (−0.994 + 0.104i)5-s + (−0.994 − 0.104i)6-s + (−0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.413 + 0.459i)11-s + (0.669 + 0.743i)12-s + (1.20 − 1.08i)13-s + (0.978 + 0.207i)14-s + (−0.743 + 0.669i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.851+0.523i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.851+0.523i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −0.851+0.523i-0.851 + 0.523i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(931,⋅)\chi_{1800} (931, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :0), −0.851+0.523i)(2,\ 1800,\ (\ :0),\ -0.851 + 0.523i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58871260090.5887126009
L(12)L(\frac12) ≈\approx 0.58871260090.5887126009
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+(0.743+0.669i)T 1 + (0.743 + 0.669i)T
3 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
5 1+(0.994−0.104i)T 1 + (0.994 - 0.104i)T
good7 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
11 1+(0.413−0.459i)T+(−0.104−0.994i)T2 1 + (0.413 - 0.459i)T + (-0.104 - 0.994i)T^{2}
13 1+(−1.20+1.08i)T+(0.104−0.994i)T2 1 + (-1.20 + 1.08i)T + (0.104 - 0.994i)T^{2}
17 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
19 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
23 1+(−0.207+0.978i)T+(−0.913−0.406i)T2 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2}
29 1+(−0.251+0.564i)T+(−0.669−0.743i)T2 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2}
31 1+(−0.406−0.913i)T+(−0.669+0.743i)T2 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2}
37 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
41 1+(1.08+1.20i)T+(−0.104+0.994i)T2 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.251−0.564i)T+(−0.669−0.743i)T2 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2}
53 1+(0.951+1.30i)T+(−0.309+0.951i)T2 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2}
59 1+(−0.104+0.994i)T2 1 + (-0.104 + 0.994i)T^{2}
61 1+(0.743+0.669i)T+(0.104+0.994i)T2 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2}
67 1+(−1.47+0.658i)T+(0.669−0.743i)T2 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2}
71 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.251−0.564i)T+(−0.669−0.743i)T2 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2}
83 1+(0.0646−0.614i)T+(−0.978−0.207i)T2 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2}
89 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
97 1+(0.669+0.743i)T2 1 + (0.669 + 0.743i)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

−8.863424082088983682809144353771, −8.506232029666333233916883038918, −7.889704196978169466386638721286, −6.86237847842204951730334455405, −6.49994868894441304259862104899, −4.73264488966425269157801326105, −3.61754481188853586988950060135, −3.03454148459537247483207469615, −2.19354483635704203801513653927, −0.50985980137464453530380470028, 1.61922432505441860853942633956, 3.17286063528004763782203761606, 4.00361320507803825237374295870, 4.72210176711896253924244599698, 6.10057884301191050481513746300, 6.73851104267984359919627495643, 7.65233953762706411129931400970, 8.374778000786076228579629624974, 8.785136106852429537099883424854, 9.589962371849454431827370334056

Graph of the ZZ-function along the critical line