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Properties

Label 2-2016-224.181-c0-0-1
Degree 22
Conductor 20162016
Sign 0.980−0.195i0.980 - 0.195i
Analytic cond. 1.006111.00611
Root an. cond. 1.003051.00305
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (1.70 + 0.707i)11-s + 1.00·14-s − 1.00·16-s + (−1.70 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−1.70 + 0.707i)29-s + (0.707 − 0.707i)32-s + (0.707 − 1.70i)37-s + (1.70 + 0.707i)43-s + (0.707 − 1.70i)44-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (1.70 + 0.707i)11-s + 1.00·14-s − 1.00·16-s + (−1.70 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−1.70 + 0.707i)29-s + (0.707 − 0.707i)32-s + (0.707 − 1.70i)37-s + (1.70 + 0.707i)43-s + (0.707 − 1.70i)44-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s) L(s)=((0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.980−0.195i0.980 - 0.195i
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2016(181,⋅)\chi_{2016} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :0), 0.980−0.195i)(2,\ 2016,\ (\ :0),\ 0.980 - 0.195i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.81377539780.8137753978
L(12)L(\frac12) ≈\approx 0.81377539780.8137753978
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1 1
7 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good5 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
11 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
13 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
23 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
29 1+(1.70−0.707i)T+(0.707−0.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.707−0.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
59 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
61 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
67 1+(−1.70+0.707i)T+(0.707−0.707i)T2 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2}
71 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
73 1+iT2 1 + iT^{2}
79 1+1.41iT−T2 1 + 1.41iT - T^{2}
83 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1−T2 1 - 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

−9.407451932946569548496525650909, −8.733814121242069168788855298411, −7.60282735797675923994024290716, −7.05697341953273384571157184019, −6.45009743160304859862112054803, −5.67856037791634364500716182528, −4.43429246972704737324941216946, −3.82059493362408161803416664553, −2.21585999056885538858314932658, −0.935488014515606690582974486929, 1.16715124412542896136250607404, 2.34618061159598827753076317807, 3.49158128908759711168418783026, 3.90452363955520181326401696080, 5.39629206560249034541068808492, 6.29768459623533328924212306573, 7.04382689776782610169229323416, 7.948744555697086866181598103708, 8.864345392298610635473553447160, 9.395353912882291639610906360418

Graph of the ZZ-function along the critical line