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Properties

Label 2-936-117.31-c0-0-1
Degree 22
Conductor 936936
Sign −0.311+0.950i-0.311 + 0.950i
Analytic cond. 0.4671240.467124
Root an. cond. 0.6834650.683465
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  + (0.5 − 0.866i)3-s + (−0.366 − 1.36i)5-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1.36 − 0.366i)15-s − i·17-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s − 0.999·27-s + (0.366 + 1.36i)31-s + (−1 − i)37-s + (0.499 + 0.866i)39-s + (−0.366 − 1.36i)41-s + (0.866 − 0.5i)43-s + (−0.999 + 0.999i)45-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.366 − 1.36i)5-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1.36 − 0.366i)15-s − i·17-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s − 0.999·27-s + (0.366 + 1.36i)31-s + (−1 − i)37-s + (0.499 + 0.866i)39-s + (−0.366 − 1.36i)41-s + (0.866 − 0.5i)43-s + (−0.999 + 0.999i)45-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.311+0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s) L(s)=((−0.311+0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.311+0.950i-0.311 + 0.950i
Analytic conductor: 0.4671240.467124
Root analytic conductor: 0.6834650.683465
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ936(265,⋅)\chi_{936} (265, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :0), −0.311+0.950i)(2,\ 936,\ (\ :0),\ -0.311 + 0.950i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0421692581.042169258
L(12)L(\frac12) ≈\approx 1.0421692581.042169258
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 1
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
7 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
11 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1−iT2 1 - iT^{2}
23 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
37 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
41 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
43 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
47 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
53 1−T+T2 1 - T + T^{2}
59 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
89 1+iT2 1 + iT^{2}
97 1+(0.866+0.5i)T2 1 + (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

−9.634889844275284843916662474084, −8.946806901484747744251020316446, −8.538787504262126513338558787988, −7.38009908491076695743494678413, −6.95804925045234238036711657787, −5.56208790554890446908582109494, −4.76371234894154358788424743507, −3.63989589581498703835923650270, −2.29472328275643872310291942626, −1.01568278684770457234785137423, 2.43725793796328625110298629381, 3.20948182947219682025111276881, 4.07174289346193477666585577473, 5.19548002672924797710506199922, 6.26531361749156595718395068163, 7.24985717774897542699329469432, 8.040729084737041350320970556565, 8.802833819231850772555676477298, 10.06901367579963915532553857959, 10.28262601030309020956755927548

Graph of the ZZ-function along the critical line