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Properties

Label 2-2016-56.13-c0-0-0
Degree 22
Conductor 20162016
Sign −i-i
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  − 7-s + 2i·11-s − 25-s + 2i·29-s + 49-s + 2i·53-s − 2i·77-s + 2·79-s − 2i·107-s + ⋯
L(s)  = 1  − 7-s + 2i·11-s − 25-s + 2i·29-s + 49-s + 2i·53-s − 2i·77-s + 2·79-s − 2i·107-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: −i-i
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2016(433,⋅)\chi_{2016} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :0), −i)(2,\ 2016,\ (\ :0),\ -i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.82532963090.8253296309
L(12)L(\frac12) ≈\approx 0.82532963090.8253296309
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+T 1 + T
good5 1+T2 1 + T^{2}
11 1−2iT−T2 1 - 2iT - T^{2}
13 1+T2 1 + T^{2}
17 1−T2 1 - T^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1−2iT−T2 1 - 2iT - T^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 1−T2 1 - T^{2}
71 1+T2 1 + T^{2}
73 1−T2 1 - T^{2}
79 1−2T+T2 1 - 2T + T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - 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

−9.495310512200955285690241385389, −9.014765985635509530895120500123, −7.80306595517540421809179504285, −7.14006411746874194488288859304, −6.55357011788315050274679739283, −5.53941744994314085129589988784, −4.63944256059288244050083158848, −3.79327996602954626710338760874, −2.73611361457109315837454974343, −1.66556939907532007794522363574, 0.58263347340642125748447033325, 2.35225331424319618699330017654, 3.36931326510326565572313214757, 3.95556161752243989093016531374, 5.33046400505539845807778232187, 6.09360191673279763545066861955, 6.52773866848835795180167618863, 7.74788973017738012710123004685, 8.337619330154305861744496684188, 9.184846187043044996609270817019

Graph of the ZZ-function along the critical line