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Properties

Label 2-2700-540.167-c0-0-1
Degree 22
Conductor 27002700
Sign 0.835+0.550i0.835 + 0.550i
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  + (0.996 + 0.0871i)2-s + (0.422 − 0.906i)3-s + (0.984 + 0.173i)4-s + (0.5 − 0.866i)6-s + (0.560 + 0.392i)7-s + (0.965 + 0.258i)8-s + (−0.642 − 0.766i)9-s + (0.573 − 0.819i)12-s + (0.524 + 0.439i)14-s + (0.939 + 0.342i)16-s + (−0.573 − 0.819i)18-s + (0.592 − 0.342i)21-s + (−0.199 − 0.284i)23-s + (0.642 − 0.766i)24-s + (−0.965 + 0.258i)27-s + (0.483 + 0.483i)28-s + ⋯
L(s)  = 1  + (0.996 + 0.0871i)2-s + (0.422 − 0.906i)3-s + (0.984 + 0.173i)4-s + (0.5 − 0.866i)6-s + (0.560 + 0.392i)7-s + (0.965 + 0.258i)8-s + (−0.642 − 0.766i)9-s + (0.573 − 0.819i)12-s + (0.524 + 0.439i)14-s + (0.939 + 0.342i)16-s + (−0.573 − 0.819i)18-s + (0.592 − 0.342i)21-s + (−0.199 − 0.284i)23-s + (0.642 − 0.766i)24-s + (−0.965 + 0.258i)27-s + (0.483 + 0.483i)28-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=((0.835+0.550i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s) L(s)=((0.835+0.550i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: 0.835+0.550i0.835 + 0.550i
Analytic conductor: 1.347471.34747
Root analytic conductor: 1.160801.16080
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2700(707,⋅)\chi_{2700} (707, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :0), 0.835+0.550i)(2,\ 2700,\ (\ :0),\ 0.835 + 0.550i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.7578147082.757814708
L(12)L(\frac12) ≈\approx 2.7578147082.757814708
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+(−0.996−0.0871i)T 1 + (-0.996 - 0.0871i)T
3 1+(−0.422+0.906i)T 1 + (-0.422 + 0.906i)T
5 1 1
good7 1+(−0.560−0.392i)T+(0.342+0.939i)T2 1 + (-0.560 - 0.392i)T + (0.342 + 0.939i)T^{2}
11 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
13 1+(0.984−0.173i)T2 1 + (0.984 - 0.173i)T^{2}
17 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+(0.199+0.284i)T+(−0.342+0.939i)T2 1 + (0.199 + 0.284i)T + (-0.342 + 0.939i)T^{2}
29 1+(1.50−1.26i)T+(0.173−0.984i)T2 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2}
31 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
37 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
41 1+(0.826−0.984i)T+(−0.173−0.984i)T2 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2}
43 1+(0.731+1.56i)T+(−0.642+0.766i)T2 1 + (0.731 + 1.56i)T + (-0.642 + 0.766i)T^{2}
47 1+(−0.878+1.25i)T+(−0.342−0.939i)T2 1 + (-0.878 + 1.25i)T + (-0.342 - 0.939i)T^{2}
53 1+iT2 1 + iT^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2}
67 1+(−1.28+0.112i)T+(0.984−0.173i)T2 1 + (-1.28 + 0.112i)T + (0.984 - 0.173i)T^{2}
71 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
73 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
79 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
83 1+(0.133−1.52i)T+(−0.984−0.173i)T2 1 + (0.133 - 1.52i)T + (-0.984 - 0.173i)T^{2}
89 1+(0.642+1.11i)T+(−0.5+0.866i)T2 1 + (0.642 + 1.11i)T + (-0.5 + 0.866i)T^{2}
97 1+(−0.642+0.766i)T2 1 + (-0.642 + 0.766i)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

−8.548641795500385332345095119113, −8.202677258512160889230585516291, −7.04501743411617800552510362328, −6.92976337982003637841915980402, −5.66350526126018542857815450556, −5.32329189986284556445632331175, −4.08320421530801451316925364150, −3.27432144532511703535870885137, −2.29015137318164710041831128273, −1.54737301774132153295091990903, 1.73006288381997480300862381656, 2.71421113240285438644760109942, 3.70161184413184259490678739798, 4.24929176231564620042439479832, 5.06461624157069535937422452383, 5.71821537405312765734417916364, 6.66200538018671417003450419465, 7.75250440272465322918617875480, 8.054375773608060400033783635337, 9.309296817579175321593277974393

Graph of the ZZ-function along the critical line