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Properties

Label 2-1800-1800.1771-c0-0-1
Degree 22
Conductor 18001800
Sign 0.851+0.523i0.851 + 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+0.523i0.851 + 0.523i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(1771,⋅)\chi_{1800} (1771, \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 2.4741297542.474129754
L(12)L(\frac12) ≈\approx 2.4741297542.474129754
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

−9.733808949505023707766585217703, −8.575349034277811693470364086324, −8.227844445742367188647444156194, −6.84781334059969072738753144786, −5.83392625493794589668403428405, −5.09939003724839557032451518265, −4.55770667521927088884364107667, −3.30908475819232154744384149184, −2.42480169018687278073725286970, −1.88987958551792556223339476912, 2.03877755983766968420454142896, 2.34862008093504993769610103878, 3.85387529982627194594816727092, 4.71532494135022391380194252583, 5.38258117509407176660646823546, 6.61379621283506029710969383790, 7.12204618024458327440316944346, 7.65281266591967465917900704997, 8.761160904402304117621828032440, 9.166598279553728271503218639841

Graph of the ZZ-function along the critical line