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Properties

Label 2-720-180.139-c0-0-1
Degree 22
Conductor 720720
Sign 0.939+0.342i0.939 + 0.342i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
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  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s − 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + (0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s − 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + (0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.939+0.342i0.939 + 0.342i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ720(319,⋅)\chi_{720} (319, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :0), 0.939+0.342i)(2,\ 720,\ (\ :0),\ 0.939 + 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2348305241.234830524
L(12)L(\frac12) ≈\approx 1.2348305241.234830524
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
29 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
47 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)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)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
89 1+T+T2 1 + T + 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

−10.20029961747774673707224432113, −9.755916473043659594658264836448, −9.048397663310796971925852685668, −8.010195232001580241879209933299, −7.28891948476599430707801398791, −6.14834496880798090102101709635, −4.89485694381613977259459875115, −4.03758408865111910624035305607, −3.15618045760290732572839988046, −1.51478032317578336594166757865, 2.24611297220220331476510645754, 2.70293794684888481033897050418, 3.87276349712210614788700350242, 5.61399028497859415870112415434, 6.36866776584097815883184440209, 7.02585684708879847186149528269, 8.270797762792726198707625039532, 8.892380100205280971221178674388, 9.766825487891479562477886837017, 10.35522067596290242243292923170

Graph of the ZZ-function along the critical line