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Properties

Label 2-1800-1800.571-c0-0-0
Degree 22
Conductor 18001800
Sign −0.979−0.201i-0.979 - 0.201i
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  + (0.207 + 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (−0.406 − 0.913i)5-s + (−0.406 + 0.913i)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 + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (−0.336 + 1.58i)13-s + (−0.669 − 0.743i)14-s + (0.207 − 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (−0.406 − 0.913i)5-s + (−0.406 + 0.913i)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 + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (−0.336 + 1.58i)13-s + (−0.669 − 0.743i)14-s + (0.207 − 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.979−0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.979−0.201i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −0.979−0.201i-0.979 - 0.201i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(571,⋅)\chi_{1800} (571, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :0), −0.979−0.201i)(2,\ 1800,\ (\ :0),\ -0.979 - 0.201i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0476613091.047661309
L(12)L(\frac12) ≈\approx 1.0476613091.047661309
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+(−0.207−0.978i)T 1 + (-0.207 - 0.978i)T
3 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
5 1+(0.406+0.913i)T 1 + (0.406 + 0.913i)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+(−0.604+0.128i)T+(0.913−0.406i)T2 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2}
13 1+(0.336−1.58i)T+(−0.913−0.406i)T2 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)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+(0.743−0.669i)T+(0.104−0.994i)T2 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2}
29 1+(−0.614−0.0646i)T+(0.978+0.207i)T2 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2}
31 1+(−0.994+0.104i)T+(0.978−0.207i)T2 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2}
37 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
41 1+(−1.58−0.336i)T+(0.913+0.406i)T2 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)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+(0.614+0.0646i)T+(0.978+0.207i)T2 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)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+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{2}
61 1+(−0.207−0.978i)T+(−0.913+0.406i)T2 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2}
67 1+(0.169+1.60i)T+(−0.978+0.207i)T2 1 + (0.169 + 1.60i)T + (-0.978 + 0.207i)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+(0.614+0.0646i)T+(0.978+0.207i)T2 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)T^{2}
83 1+(−0.564−0.251i)T+(0.669+0.743i)T2 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)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+(−0.978−0.207i)T2 1 + (-0.978 - 0.207i)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

−9.503170520082510548853970251582, −8.810009667984859065319476163759, −8.524023452907433907597619345052, −7.57081672821671703803960974335, −6.59389886971813200037069872030, −5.93843669275691616154422410598, −4.70301509100674632861968531469, −4.20578577615222657163917852115, −3.51048013425426363447693155273, −2.02193224799647274025422948101, 0.65787479445534311836857715219, 2.45751851769454879973491021913, 2.86963483958751385997363460974, 3.78709737255910691061524252919, 4.53954607183980206653741083883, 6.13871269550395831840598914747, 6.67200840912887389296683816181, 7.62497057433924395507930582390, 8.377618318535007875934857708536, 9.213045177873701217670138299007

Graph of the ZZ-function along the critical line