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Properties

Label 2-728-728.237-c0-0-3
Degree 22
Conductor 728728
Sign 0.967+0.252i0.967 + 0.252i
Analytic cond. 0.3633190.363319
Root an. cond. 0.6027590.602759
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 + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 − 1.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.49i)10-s + 1.73·12-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−1.49 − 2.59i)15-s + (−0.5 − 0.866i)16-s − 2·18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 − 1.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.49i)10-s + 1.73·12-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−1.49 − 2.59i)15-s + (−0.5 − 0.866i)16-s − 2·18-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((0.967+0.252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s) L(s)=((0.967+0.252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 0.967+0.252i0.967 + 0.252i
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ728(237,⋅)\chi_{728} (237, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :0), 0.967+0.252i)(2,\ 728,\ (\ :0),\ 0.967 + 0.252i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1350949351.135094935
L(12)L(\frac12) ≈\approx 1.1350949351.135094935
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
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good3 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
5 1−1.73T+T2 1 - 1.73T + T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
73 1−T2 1 - T^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−10.63157422720753413224646223117, −9.704823732937147800103367237743, −8.548102689221723831599473891750, −7.37503432491491403919223077770, −7.13135071590034197549836871651, −6.04670808622422837535819918405, −5.57357652766491961380423211411, −4.72196742669697703902744266762, −2.70531240019805837894332022205, −1.40302546205491119264690240160, 1.94224099105061124587509288696, 3.01380030288860445188221829542, 4.59982418802210044406978793320, 5.02074188124954739642311961318, 5.79829003412315497443300531174, 6.44955491513201741717966213335, 8.774809372988672521730911430359, 9.414234912420366850561707505944, 9.871378168251772001822184329977, 10.63868279463485881251013491010

Graph of the ZZ-function along the critical line