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Properties

Label 2-936-936.571-c0-0-5
Degree 22
Conductor 936936
Sign 0.766+0.642i0.766 + 0.642i
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)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (0.173 − 0.300i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 1.62i)7-s − 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 1.62i)14-s + (−0.266 − 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (−0.173 − 0.984i)18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (0.173 − 0.300i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 1.62i)7-s − 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 1.62i)14-s + (−0.266 − 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (−0.173 − 0.984i)18-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.766+0.642i0.766 + 0.642i
Analytic conductor: 0.4671240.467124
Root analytic conductor: 0.6834650.683465
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ936(571,⋅)\chi_{936} (571, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :0), 0.766+0.642i)(2,\ 936,\ (\ :0),\ 0.766 + 0.642i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1454993861.145499386
L(12)L(\frac12) ≈\approx 1.1454993861.145499386
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
13 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(−0.173+0.300i)T+(−0.5−0.866i)T2 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2}
7 1+(0.939+1.62i)T+(−0.5+0.866i)T2 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1−1.53T+T2 1 - 1.53T + T^{2}
19 1−T2 1 - T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+0.347T+T2 1 + 0.347T + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
47 1+(−0.766−1.32i)T+(−0.5+0.866i)T2 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1−1.87T+T2 1 - 1.87T + T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−10.06506900870705766951354793276, −9.110145679098978791874479698929, −8.135368004430734219788789463738, −7.43885934550884356311291129598, −6.89903214080491936996079729513, −5.99532101080813594931681917486, −5.18244894728401172227236820279, −3.67665001129766546784627538157, −3.18856189737395708491296367380, −0.992581125609915865638302776907, 2.19880977026496456782097009518, 3.07314402366013536093382312096, 3.81801845966138062638154493539, 5.07795930920553080306164077079, 5.78702037269898557515899124224, 6.46894571734855944864049408598, 8.314646689570212762592143230686, 9.069243256547853761477605954659, 9.669485139608179540860869964529, 10.21133396582938225634760294742

Graph of the ZZ-function along the critical line