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Properties

Label 2-1800-360.347-c0-0-2
Degree 22
Conductor 18001800
Sign 0.116−0.993i0.116 - 0.993i
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  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 + 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)18-s + i·19-s + (1.67 − 0.448i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 + 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)18-s + i·19-s + (1.67 − 0.448i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((0.116−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s) L(s)=((0.116−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 0.116−0.993i0.116 - 0.993i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(707,⋅)\chi_{1800} (707, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :0), 0.116−0.993i)(2,\ 1800,\ (\ :0),\ 0.116 - 0.993i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9689564101.968956410
L(12)L(\frac12) ≈\approx 1.9689564101.968956410
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+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
3 1+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
5 1 1
good7 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
11 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
13 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
17 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
19 1−iT−T2 1 - iT - T^{2}
23 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−iT2 1 - iT^{2}
41 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
43 1+(1.67+0.448i)T+(0.866+0.5i)T2 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2}
47 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
53 1−iT2 1 - iT^{2}
59 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−1.67+0.448i)T+(0.866−0.5i)T2 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2}
71 1+T2 1 + T^{2}
73 1+(1.22−1.22i)T−iT2 1 + (1.22 - 1.22i)T - iT^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1+(−0.517+1.93i)T+(−0.866−0.5i)T2 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.448−1.67i)T+(−0.866−0.5i)T2 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)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

−9.778480808026806109247240786824, −8.643908240725626686218500653805, −8.316937788372193961742678233962, −6.83164014416569120514138836681, −6.32625651852690267667933217092, −5.55914196292768435859230051155, −4.70841884075712064905390133552, −3.68637544252553756347205513559, −3.49121543569572155018620683639, −1.84310965810612597520566317995, 1.31007127493659229095360362619, 2.24288356610607951842524322984, 3.28331328081662677597770206235, 4.48518277291294117730704224536, 5.09005332031294500503907237940, 6.25027319000922646611968505345, 6.80913056421222819378205074131, 7.22025309703834028010800342997, 8.412345208350741381674378660192, 9.349725960141697994451509073479

Graph of the ZZ-function along the critical line