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Properties

Label 2-161-1.1-c3-0-2
Degree 22
Conductor 161161
Sign 11
Analytic cond. 9.499309.49930
Root an. cond. 3.082093.08209
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.60·2-s − 7.98·3-s + 23.4·4-s − 5.27·5-s + 44.7·6-s + 7·7-s − 86.7·8-s + 36.7·9-s + 29.5·10-s − 50.0·11-s − 187.·12-s − 60.3·13-s − 39.2·14-s + 42.0·15-s + 298.·16-s − 87.5·17-s − 206.·18-s − 23.8·19-s − 123.·20-s − 55.8·21-s + 281.·22-s + 23·23-s + 692.·24-s − 97.2·25-s + 338.·26-s − 77.8·27-s + 164.·28-s + ⋯
L(s)  = 1  − 1.98·2-s − 1.53·3-s + 2.93·4-s − 0.471·5-s + 3.04·6-s + 0.377·7-s − 3.83·8-s + 1.36·9-s + 0.934·10-s − 1.37·11-s − 4.50·12-s − 1.28·13-s − 0.749·14-s + 0.724·15-s + 4.67·16-s − 1.24·17-s − 2.69·18-s − 0.287·19-s − 1.38·20-s − 0.580·21-s + 2.72·22-s + 0.208·23-s + 5.89·24-s − 0.777·25-s + 2.55·26-s − 0.554·27-s + 1.10·28-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 11
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), 1)(2,\ 161,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.13054317160.1305431716
L(12)L(\frac12) ≈\approx 0.13054317160.1305431716
L(52)L(\frac{5}{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)
bad7 1−7T 1 - 7T
23 1−23T 1 - 23T
good2 1+5.60T+8T2 1 + 5.60T + 8T^{2}
3 1+7.98T+27T2 1 + 7.98T + 27T^{2}
5 1+5.27T+125T2 1 + 5.27T + 125T^{2}
11 1+50.0T+1.33e3T2 1 + 50.0T + 1.33e3T^{2}
13 1+60.3T+2.19e3T2 1 + 60.3T + 2.19e3T^{2}
17 1+87.5T+4.91e3T2 1 + 87.5T + 4.91e3T^{2}
19 1+23.8T+6.85e3T2 1 + 23.8T + 6.85e3T^{2}
29 1−102.T+2.43e4T2 1 - 102.T + 2.43e4T^{2}
31 1+118.T+2.97e4T2 1 + 118.T + 2.97e4T^{2}
37 1−215.T+5.06e4T2 1 - 215.T + 5.06e4T^{2}
41 1−43.5T+6.89e4T2 1 - 43.5T + 6.89e4T^{2}
43 1+165.T+7.95e4T2 1 + 165.T + 7.95e4T^{2}
47 1+54.1T+1.03e5T2 1 + 54.1T + 1.03e5T^{2}
53 1+228.T+1.48e5T2 1 + 228.T + 1.48e5T^{2}
59 1−606.T+2.05e5T2 1 - 606.T + 2.05e5T^{2}
61 1+752.T+2.26e5T2 1 + 752.T + 2.26e5T^{2}
67 1−358.T+3.00e5T2 1 - 358.T + 3.00e5T^{2}
71 1−699.T+3.57e5T2 1 - 699.T + 3.57e5T^{2}
73 1−255.T+3.89e5T2 1 - 255.T + 3.89e5T^{2}
79 1−1.32e3T+4.93e5T2 1 - 1.32e3T + 4.93e5T^{2}
83 1+43.6T+5.71e5T2 1 + 43.6T + 5.71e5T^{2}
89 1−896.T+7.04e5T2 1 - 896.T + 7.04e5T^{2}
97 1−56.3T+9.12e5T2 1 - 56.3T + 9.12e5T^{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

−11.84127664173250162531695891730, −11.08834025310922745318875879325, −10.52341721369445726007278311689, −9.560670816209608978735223614493, −8.167689947374689944652251870027, −7.36008133627449215169925413512, −6.37617335821426044812059421537, −5.08659949497351894994538905413, −2.29320278579173792343394336858, −0.38935439294581919508885722813, 0.38935439294581919508885722813, 2.29320278579173792343394336858, 5.08659949497351894994538905413, 6.37617335821426044812059421537, 7.36008133627449215169925413512, 8.167689947374689944652251870027, 9.560670816209608978735223614493, 10.52341721369445726007278311689, 11.08834025310922745318875879325, 11.84127664173250162531695891730

Graph of the ZZ-function along the critical line