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Properties

Label 2-2700-108.67-c0-0-1
Degree 22
Conductor 27002700
Sign −0.286+0.957i-0.286 + 0.957i
Analytic cond. 1.347471.34747
Root an. cond. 1.160801.16080
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  + (−0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (0.524 − 1.43i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (−0.266 + 1.50i)14-s + (0.766 − 0.642i)16-s + (0.342 + 0.939i)18-s + (−0.766 − 1.32i)21-s + (−0.642 − 1.76i)23-s + (−0.173 + 0.984i)24-s + (−0.866 − 0.500i)27-s − 1.53i·28-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (0.524 − 1.43i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (−0.266 + 1.50i)14-s + (0.766 − 0.642i)16-s + (0.342 + 0.939i)18-s + (−0.766 − 1.32i)21-s + (−0.642 − 1.76i)23-s + (−0.173 + 0.984i)24-s + (−0.866 − 0.500i)27-s − 1.53i·28-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=((−0.286+0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s) L(s)=((−0.286+0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: −0.286+0.957i-0.286 + 0.957i
Analytic conductor: 1.347471.34747
Root analytic conductor: 1.160801.16080
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2700(2551,⋅)\chi_{2700} (2551, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :0), −0.286+0.957i)(2,\ 2700,\ (\ :0),\ -0.286 + 0.957i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97830975190.9783097519
L(12)L(\frac12) ≈\approx 0.97830975190.9783097519
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+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
3 1+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
5 1 1
good7 1+(−0.524+1.43i)T+(−0.766−0.642i)T2 1 + (-0.524 + 1.43i)T + (-0.766 - 0.642i)T^{2}
11 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
13 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.642+1.76i)T+(−0.766+0.642i)T2 1 + (0.642 + 1.76i)T + (-0.766 + 0.642i)T^{2}
29 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2}
31 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
37 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
41 1+(−0.0603+0.342i)T+(−0.939−0.342i)T2 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2}
43 1+(−0.642−0.766i)T+(−0.173+0.984i)T2 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2}
47 1+(−0.118+0.326i)T+(−0.766−0.642i)T2 1 + (-0.118 + 0.326i)T + (-0.766 - 0.642i)T^{2}
53 1+T2 1 + T^{2}
59 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
61 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
67 1+(−0.342−0.0603i)T+(0.939+0.342i)T2 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
79 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
83 1+(−0.342+0.0603i)T+(0.939−0.342i)T2 1 + (-0.342 + 0.0603i)T + (0.939 - 0.342i)T^{2}
89 1+(−0.173−0.300i)T+(−0.5+0.866i)T2 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2}
97 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)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

−8.668215478462960942155954387969, −7.987292492760681331729078467521, −7.45506660597276357904128138323, −6.78327496076935701114857377226, −6.21443983189667071583706754333, −4.89219198762681392258003728802, −3.81747760709103412527986523361, −2.79181619916161648210137172965, −1.73461319208688640665134456575, −0.808289401177368800397972528983, 1.79392496398567069569763812558, 2.48359538950325078480539953720, 3.36504854255841829321577638191, 4.40859098661331588517012113835, 5.55988879020140216667723760034, 6.04480208281299186987125181893, 7.42472366617588390356882833819, 7.990479821873603614011478759010, 8.594215582176858476247169049888, 9.288153656828913580462220028896

Graph of the ZZ-function along the critical line