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Properties

Label 2-162-1.1-c7-0-15
Degree 22
Conductor 162162
Sign 11
Analytic cond. 50.606350.6063
Root an. cond. 7.113817.11381
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 64·4-s + 320.·5-s + 1.19e3·7-s − 512·8-s − 2.56e3·10-s + 6.80e3·11-s + 7.47e3·13-s − 9.55e3·14-s + 4.09e3·16-s + 3.45e4·17-s + 1.79e4·19-s + 2.05e4·20-s − 5.44e4·22-s − 1.70e4·23-s + 2.46e4·25-s − 5.97e4·26-s + 7.64e4·28-s − 7.39e4·29-s − 5.16e4·31-s − 3.27e4·32-s − 2.76e5·34-s + 3.82e5·35-s − 5.17e5·37-s − 1.43e5·38-s − 1.64e5·40-s + 2.03e5·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.14·5-s + 1.31·7-s − 0.353·8-s − 0.810·10-s + 1.54·11-s + 0.943·13-s − 0.930·14-s + 0.250·16-s + 1.70·17-s + 0.601·19-s + 0.573·20-s − 1.09·22-s − 0.291·23-s + 0.315·25-s − 0.667·26-s + 0.657·28-s − 0.562·29-s − 0.311·31-s − 0.176·32-s − 1.20·34-s + 1.50·35-s − 1.68·37-s − 0.425·38-s − 0.405·40-s + 0.460·41-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 11
Analytic conductor: 50.606350.6063
Root analytic conductor: 7.113817.11381
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 162, ( :7/2), 1)(2,\ 162,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.7720627612.772062761
L(12)L(\frac12) ≈\approx 2.7720627612.772062761
L(92)L(\frac{9}{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+8T 1 + 8T
3 1 1
good5 1−320.T+7.81e4T2 1 - 320.T + 7.81e4T^{2}
7 1−1.19e3T+8.23e5T2 1 - 1.19e3T + 8.23e5T^{2}
11 1−6.80e3T+1.94e7T2 1 - 6.80e3T + 1.94e7T^{2}
13 1−7.47e3T+6.27e7T2 1 - 7.47e3T + 6.27e7T^{2}
17 1−3.45e4T+4.10e8T2 1 - 3.45e4T + 4.10e8T^{2}
19 1−1.79e4T+8.93e8T2 1 - 1.79e4T + 8.93e8T^{2}
23 1+1.70e4T+3.40e9T2 1 + 1.70e4T + 3.40e9T^{2}
29 1+7.39e4T+1.72e10T2 1 + 7.39e4T + 1.72e10T^{2}
31 1+5.16e4T+2.75e10T2 1 + 5.16e4T + 2.75e10T^{2}
37 1+5.17e5T+9.49e10T2 1 + 5.17e5T + 9.49e10T^{2}
41 1−2.03e5T+1.94e11T2 1 - 2.03e5T + 1.94e11T^{2}
43 1+9.73e5T+2.71e11T2 1 + 9.73e5T + 2.71e11T^{2}
47 1−7.37e5T+5.06e11T2 1 - 7.37e5T + 5.06e11T^{2}
53 1−5.08e5T+1.17e12T2 1 - 5.08e5T + 1.17e12T^{2}
59 1+1.90e6T+2.48e12T2 1 + 1.90e6T + 2.48e12T^{2}
61 1+2.07e6T+3.14e12T2 1 + 2.07e6T + 3.14e12T^{2}
67 1+2.98e6T+6.06e12T2 1 + 2.98e6T + 6.06e12T^{2}
71 1−2.98e6T+9.09e12T2 1 - 2.98e6T + 9.09e12T^{2}
73 1−5.26e6T+1.10e13T2 1 - 5.26e6T + 1.10e13T^{2}
79 1+1.39e6T+1.92e13T2 1 + 1.39e6T + 1.92e13T^{2}
83 1+5.23e5T+2.71e13T2 1 + 5.23e5T + 2.71e13T^{2}
89 1+4.57e6T+4.42e13T2 1 + 4.57e6T + 4.42e13T^{2}
97 1+1.02e7T+8.07e13T2 1 + 1.02e7T + 8.07e13T^{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

−11.45570590780023874320557567955, −10.41021592605496545386155357545, −9.464499930887899250102294880192, −8.622916873325483920549799157987, −7.52502321592806223024002675380, −6.22930630098161077520130002615, −5.30663453183460788678526653635, −3.57807503794106469640820012141, −1.69357054407533087992533680894, −1.26078895091058140871697380014, 1.26078895091058140871697380014, 1.69357054407533087992533680894, 3.57807503794106469640820012141, 5.30663453183460788678526653635, 6.22930630098161077520130002615, 7.52502321592806223024002675380, 8.622916873325483920549799157987, 9.464499930887899250102294880192, 10.41021592605496545386155357545, 11.45570590780023874320557567955

Graph of the ZZ-function along the critical line