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Properties

Label 2-2016-56.13-c0-0-1
Degree 22
Conductor 20162016
Sign 11
Analytic cond. 1.006111.00611
Root an. cond. 1.003051.00305
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 2·23-s − 25-s + 49-s − 2·71-s + 2·79-s + 2·113-s + ⋯
L(s)  = 1  + 7-s + 2·23-s − 25-s + 49-s − 2·71-s + 2·79-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ2016(433,⋅)\chi_{2016} (433, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2016, ( :0), 1)(2,\ 2016,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3218423951.321842395
L(12)L(\frac12) ≈\approx 1.3218423951.321842395
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−T)(1+T) ( 1 - T )( 1 + T )
13 1+T2 1 + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 1+T2 1 + T^{2}
23 (1−T)2 ( 1 - T )^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1+T)2 ( 1 + T )^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)2 ( 1 - T )^{2}
83 1+T2 1 + T^{2}
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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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

−9.196521524049055996965274831445, −8.609845861317562863991428662898, −7.73809760233912939438783173517, −7.15154253672002626812165505831, −6.13778775498451419586192957947, −5.22219344825821505019401964028, −4.59050143743240089575299917476, −3.54015364659302204675424728415, −2.41728989834666271512798557334, −1.27340330965313936869979181495, 1.27340330965313936869979181495, 2.41728989834666271512798557334, 3.54015364659302204675424728415, 4.59050143743240089575299917476, 5.22219344825821505019401964028, 6.13778775498451419586192957947, 7.15154253672002626812165505831, 7.73809760233912939438783173517, 8.609845861317562863991428662898, 9.196521524049055996965274831445

Graph of the ZZ-function along the critical line