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Properties

Label 2-308-308.307-c0-0-1
Degree 22
Conductor 308308
Sign 11
Analytic cond. 0.1537120.153712
Root an. cond. 0.3920610.392061
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  + 2-s + 4-s − 7-s + 8-s − 9-s − 11-s − 14-s + 16-s − 18-s − 22-s + 25-s − 28-s + 32-s − 36-s − 2·37-s + 2·43-s − 44-s + 49-s + 50-s − 2·53-s − 56-s + 63-s + 64-s − 72-s − 2·74-s + 77-s + 2·79-s + ⋯
L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 9-s − 11-s − 14-s + 16-s − 18-s − 22-s + 25-s − 28-s + 32-s − 36-s − 2·37-s + 2·43-s − 44-s + 49-s + 50-s − 2·53-s − 56-s + 63-s + 64-s − 72-s − 2·74-s + 77-s + 2·79-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 0.1537120.153712
Root analytic conductor: 0.3920610.392061
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ308(307,⋅)\chi_{308} (307, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 308, ( :0), 1)(2,\ 308,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1944443701.194444370
L(12)L(\frac12) ≈\approx 1.1944443701.194444370
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−T 1 - T
7 1+T 1 + T
11 1+T 1 + T
good3 1+T2 1 + T^{2}
5 (1−T)(1+T) ( 1 - T )( 1 + T )
13 1+T2 1 + T^{2}
17 1+T2 1 + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T2 1 + T^{2}
37 (1+T)2 ( 1 + T )^{2}
41 1+T2 1 + T^{2}
43 (1−T)2 ( 1 - T )^{2}
47 1+T2 1 + T^{2}
53 (1+T)2 ( 1 + T )^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1+T2 1 + T^{2}
79 (1−T)2 ( 1 - T )^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
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

−12.22444025991043263408956350708, −11.02761844551828325954706001299, −10.41138502805636840024843453249, −9.125716297240622943033830407001, −7.919956933541530537490995495877, −6.82003577140905773126479937248, −5.87497498597523816414932977436, −4.98255767812676813049156880337, −3.46742102224712460117571879696, −2.58191647953742011947022981592, 2.58191647953742011947022981592, 3.46742102224712460117571879696, 4.98255767812676813049156880337, 5.87497498597523816414932977436, 6.82003577140905773126479937248, 7.919956933541530537490995495877, 9.125716297240622943033830407001, 10.41138502805636840024843453249, 11.02761844551828325954706001299, 12.22444025991043263408956350708

Graph of the ZZ-function along the critical line