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Properties

Label 2-4536-1.1-c1-0-42
Degree 22
Conductor 45364536
Sign 11
Analytic cond. 36.220136.2201
Root an. cond. 6.018316.01831
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 6·11-s + 6·13-s + 2·17-s + 7·19-s + 23-s − 4·25-s − 2·29-s + 10·31-s + 35-s − 6·37-s + 8·41-s − 10·43-s − 8·47-s + 49-s − 2·53-s + 6·55-s + 7·61-s + 6·65-s − 12·67-s − 15·71-s − 2·73-s + 6·77-s + 79-s − 12·83-s + 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.80·11-s + 1.66·13-s + 0.485·17-s + 1.60·19-s + 0.208·23-s − 4/5·25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 0.986·37-s + 1.24·41-s − 1.52·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.809·55-s + 0.896·61-s + 0.744·65-s − 1.46·67-s − 1.78·71-s − 0.234·73-s + 0.683·77-s + 0.112·79-s − 1.31·83-s + 0.216·85-s + ⋯

Functional equation

Λ(s)=(4536s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4536s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 45364536    =    23⋅34⋅72^{3} \cdot 3^{4} \cdot 7
Sign: 11
Analytic conductor: 36.220136.2201
Root analytic conductor: 6.018316.01831
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4536, ( :1/2), 1)(2,\ 4536,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0547930343.054793034
L(12)L(\frac12) ≈\approx 3.0547930343.054793034
L(32)L(\frac{3}{2}) 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−T+pT2 1 - T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+15T+pT2 1 + 15 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+4T+pT2 1 + 4 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p 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

−8.460095941009109367604154676152, −7.62853361422202650085332123774, −6.75778009886636317888600581021, −6.13984952667230449829780014899, −5.55782610661352515541339492471, −4.53630096095552390250482974797, −3.72462631167540553827048166168, −3.08371928803854652319736627806, −1.54842755853086999230961203081, −1.18913645458013163821193224515, 1.18913645458013163821193224515, 1.54842755853086999230961203081, 3.08371928803854652319736627806, 3.72462631167540553827048166168, 4.53630096095552390250482974797, 5.55782610661352515541339492471, 6.13984952667230449829780014899, 6.75778009886636317888600581021, 7.62853361422202650085332123774, 8.460095941009109367604154676152

Graph of the ZZ-function along the critical line