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Properties

Label 2-936-936.259-c0-0-3
Degree 22
Conductor 936936
Sign 0.173+0.984i0.173 + 0.984i
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 + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 − 0.984i)6-s + (0.173 − 0.300i)7-s − 0.999·8-s + (0.173 − 0.984i)9-s + 1.53·10-s + (−0.939 − 0.342i)12-s + (0.5 + 0.866i)13-s + (−0.173 − 0.300i)14-s + (1.43 + 0.524i)15-s + (−0.5 + 0.866i)16-s − 1.87·17-s + (−0.766 − 0.642i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 − 0.984i)6-s + (0.173 − 0.300i)7-s − 0.999·8-s + (0.173 − 0.984i)9-s + 1.53·10-s + (−0.939 − 0.342i)12-s + (0.5 + 0.866i)13-s + (−0.173 − 0.300i)14-s + (1.43 + 0.524i)15-s + (−0.5 + 0.866i)16-s − 1.87·17-s + (−0.766 − 0.642i)18-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.173+0.984i0.173 + 0.984i
Analytic conductor: 0.4671240.467124
Root analytic conductor: 0.6834650.683465
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ936(259,⋅)\chi_{936} (259, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :0), 0.173+0.984i)(2,\ 936,\ (\ :0),\ 0.173 + 0.984i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6690093651.669009365
L(12)L(\frac12) ≈\approx 1.6690093651.669009365
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+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
13 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(−0.766−1.32i)T+(−0.5+0.866i)T2 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}
7 1+(−0.173+0.300i)T+(−0.5−0.866i)T2 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+1.87T+T2 1 + 1.87T + 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+1.53T+T2 1 + 1.53T + T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)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+0.347T+T2 1 + 0.347T + 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

−10.23103262159873894386315506852, −9.264576352207110235489033930428, −8.761876564877716050713368707085, −7.30639573699005427488699116700, −6.59353433252468939667676209800, −5.96570693415535447880588695325, −4.40104435265045100063904526627, −3.49317464137807264296066886381, −2.42352033014093584883473888023, −1.81834806125956389067996948600, 2.06493405684043981181533529892, 3.40633640256392816317199152755, 4.54001301664613502700015127109, 5.09266895522690221936234168107, 5.89701763050257807392212344778, 7.05567871107031819010618893597, 8.291496048935321125587524812559, 8.682585368851903913511732261380, 9.189247743143536352258125867563, 10.16094061877368684262317111471

Graph of the ZZ-function along the critical line