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Properties

Label 2-162-1.1-c5-0-13
Degree 22
Conductor 162162
Sign −1-1
Analytic cond. 25.982125.9821
Root an. cond. 5.097275.09727
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 2.39·5-s − 51.6·7-s − 64·8-s + 9.57·10-s + 670.·11-s − 846.·13-s + 206.·14-s + 256·16-s − 1.13e3·17-s + 1.00e3·19-s − 38.3·20-s − 2.68e3·22-s + 2.40e3·23-s − 3.11e3·25-s + 3.38e3·26-s − 827.·28-s + 3.76e3·29-s + 4.03e3·31-s − 1.02e3·32-s + 4.54e3·34-s + 123.·35-s − 1.45e4·37-s − 4.00e3·38-s + 153.·40-s − 7.88e3·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.0428·5-s − 0.398·7-s − 0.353·8-s + 0.0302·10-s + 1.67·11-s − 1.38·13-s + 0.281·14-s + 0.250·16-s − 0.952·17-s + 0.637·19-s − 0.0214·20-s − 1.18·22-s + 0.946·23-s − 0.998·25-s + 0.982·26-s − 0.199·28-s + 0.832·29-s + 0.753·31-s − 0.176·32-s + 0.673·34-s + 0.0170·35-s − 1.74·37-s − 0.450·38-s + 0.0151·40-s − 0.732·41-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −1-1
Analytic conductor: 25.982125.9821
Root analytic conductor: 5.097275.09727
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 162, ( :5/2), −1)(2,\ 162,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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+4T 1 + 4T
3 1 1
good5 1+2.39T+3.12e3T2 1 + 2.39T + 3.12e3T^{2}
7 1+51.6T+1.68e4T2 1 + 51.6T + 1.68e4T^{2}
11 1−670.T+1.61e5T2 1 - 670.T + 1.61e5T^{2}
13 1+846.T+3.71e5T2 1 + 846.T + 3.71e5T^{2}
17 1+1.13e3T+1.41e6T2 1 + 1.13e3T + 1.41e6T^{2}
19 1−1.00e3T+2.47e6T2 1 - 1.00e3T + 2.47e6T^{2}
23 1−2.40e3T+6.43e6T2 1 - 2.40e3T + 6.43e6T^{2}
29 1−3.76e3T+2.05e7T2 1 - 3.76e3T + 2.05e7T^{2}
31 1−4.03e3T+2.86e7T2 1 - 4.03e3T + 2.86e7T^{2}
37 1+1.45e4T+6.93e7T2 1 + 1.45e4T + 6.93e7T^{2}
41 1+7.88e3T+1.15e8T2 1 + 7.88e3T + 1.15e8T^{2}
43 1+2.03e4T+1.47e8T2 1 + 2.03e4T + 1.47e8T^{2}
47 1+1.61e4T+2.29e8T2 1 + 1.61e4T + 2.29e8T^{2}
53 1+5.54e3T+4.18e8T2 1 + 5.54e3T + 4.18e8T^{2}
59 1+6.41e3T+7.14e8T2 1 + 6.41e3T + 7.14e8T^{2}
61 1−8.88e3T+8.44e8T2 1 - 8.88e3T + 8.44e8T^{2}
67 1+4.29e4T+1.35e9T2 1 + 4.29e4T + 1.35e9T^{2}
71 1+6.33e4T+1.80e9T2 1 + 6.33e4T + 1.80e9T^{2}
73 1+1.28e4T+2.07e9T2 1 + 1.28e4T + 2.07e9T^{2}
79 1+2.92e3T+3.07e9T2 1 + 2.92e3T + 3.07e9T^{2}
83 1+4.75e4T+3.93e9T2 1 + 4.75e4T + 3.93e9T^{2}
89 1−8.47e4T+5.58e9T2 1 - 8.47e4T + 5.58e9T^{2}
97 1−1.11e5T+8.58e9T2 1 - 1.11e5T + 8.58e9T^{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

−11.61298425393125072477097267809, −10.18230866603900699321846591041, −9.442414125922546825429919912384, −8.540636407442362719359062707768, −7.13806529114169311531517972716, −6.46618867659138973536813763792, −4.78352265788203075760844385039, −3.21747633950733596213687698259, −1.62572664444521784498626883956, 0, 1.62572664444521784498626883956, 3.21747633950733596213687698259, 4.78352265788203075760844385039, 6.46618867659138973536813763792, 7.13806529114169311531517972716, 8.540636407442362719359062707768, 9.442414125922546825429919912384, 10.18230866603900699321846591041, 11.61298425393125072477097267809

Graph of the ZZ-function along the critical line