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Properties

Label 2-2016-28.11-c0-0-1
Degree 22
Conductor 20162016
Sign 0.947+0.319i0.947 + 0.319i
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  + (0.5 + 0.866i)5-s − i·7-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)47-s − 49-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s − i·7-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)47-s − 49-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.947+0.319i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s) L(s)=((0.947+0.319i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.947+0.319i0.947 + 0.319i
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2016(991,⋅)\chi_{2016} (991, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :0), 0.947+0.319i)(2,\ 2016,\ (\ :0),\ 0.947 + 0.319i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2592513311.259251331
L(12)L(\frac12) ≈\approx 1.2592513311.259251331
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 1
7 1+iT 1 + iT
good5 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
19 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+T2 1 + T^{2}
31 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
37 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1−T2 1 - T^{2}
47 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
59 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (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+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{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

−9.507836971990799624024658431948, −8.432999451299898205219519363898, −7.59388142950192727698524608465, −6.99019633507703100870972428139, −6.27856565477865577395681115374, −5.25701013261298193427637008779, −4.50837949496145981223053116923, −3.11120316724633613472369783783, −2.77853052213191372413307267916, −1.05388797508579704550789810452, 1.41251267060910699789832089450, 2.42231405924243423694679105265, 3.49844705769664769310765060146, 4.79300881731443883999701089725, 5.39704011476411240567593763831, 5.93452839075905779406593191283, 7.07499284015821160075765512658, 8.068774718947576421731831577336, 8.506177687810454783606270367899, 9.611522971258847856797550952682

Graph of the ZZ-function along the critical line