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Properties

Label 2-162-1.1-c7-0-22
Degree 22
Conductor 162162
Sign −1-1
Analytic cond. 50.606350.6063
Root an. cond. 7.113817.11381
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 64·4-s − 165·5-s − 508·7-s + 512·8-s − 1.32e3·10-s + 3.02e3·11-s + 5.03e3·13-s − 4.06e3·14-s + 4.09e3·16-s − 3.18e3·17-s + 1.50e3·19-s − 1.05e4·20-s + 2.41e4·22-s − 7.56e4·23-s − 5.09e4·25-s + 4.03e4·26-s − 3.25e4·28-s − 8.26e4·29-s − 1.74e5·31-s + 3.27e4·32-s − 2.55e4·34-s + 8.38e4·35-s − 3.23e5·37-s + 1.20e4·38-s − 8.44e4·40-s − 3.08e5·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.590·5-s − 0.559·7-s + 0.353·8-s − 0.417·10-s + 0.685·11-s + 0.636·13-s − 0.395·14-s + 1/4·16-s − 0.157·17-s + 0.0504·19-s − 0.295·20-s + 0.484·22-s − 1.29·23-s − 0.651·25-s + 0.449·26-s − 0.279·28-s − 0.629·29-s − 1.05·31-s + 0.176·32-s − 0.111·34-s + 0.330·35-s − 1.05·37-s + 0.0356·38-s − 0.208·40-s − 0.698·41-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: −1-1
Analytic conductor: 50.606350.6063
Root analytic conductor: 7.113817.11381
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 162, ( :7/2), −1)(2,\ 162,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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−p3T 1 - p^{3} T
3 1 1
good5 1+33pT+p7T2 1 + 33 p T + p^{7} T^{2}
7 1+508T+p7T2 1 + 508 T + p^{7} T^{2}
11 1−3024T+p7T2 1 - 3024 T + p^{7} T^{2}
13 1−5039T+p7T2 1 - 5039 T + p^{7} T^{2}
17 1+3189T+p7T2 1 + 3189 T + p^{7} T^{2}
19 1−1508T+p7T2 1 - 1508 T + p^{7} T^{2}
23 1+75600T+p7T2 1 + 75600 T + p^{7} T^{2}
29 1+82665T+p7T2 1 + 82665 T + p^{7} T^{2}
31 1+174892T+p7T2 1 + 174892 T + p^{7} T^{2}
37 1+323569T+p7T2 1 + 323569 T + p^{7} T^{2}
41 1+308118T+p7T2 1 + 308118 T + p^{7} T^{2}
43 1−336680T+p7T2 1 - 336680 T + p^{7} T^{2}
47 1+383196T+p7T2 1 + 383196 T + p^{7} T^{2}
53 1−760206T+p7T2 1 - 760206 T + p^{7} T^{2}
59 1+2225664T+p7T2 1 + 2225664 T + p^{7} T^{2}
61 1−2244815T+p7T2 1 - 2244815 T + p^{7} T^{2}
67 1−1473188T+p7T2 1 - 1473188 T + p^{7} T^{2}
71 1+5006892T+p7T2 1 + 5006892 T + p^{7} T^{2}
73 1+5898301T+p7T2 1 + 5898301 T + p^{7} T^{2}
79 1−7028768T+p7T2 1 - 7028768 T + p^{7} T^{2}
83 1+2651196T+p7T2 1 + 2651196 T + p^{7} T^{2}
89 1+6770901T+p7T2 1 + 6770901 T + p^{7} T^{2}
97 1−16176386T+p7T2 1 - 16176386 T + p^{7} 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

−11.34043325297840783623196539298, −10.18069604097718888816905520926, −8.971694411553695824216425530445, −7.74597652412341814988104030203, −6.64173360325397727413371338916, −5.66037360169780218180545218815, −4.14021563316544553018951225009, −3.41345366030187447043885596084, −1.75635296670456043562076078548, 0, 1.75635296670456043562076078548, 3.41345366030187447043885596084, 4.14021563316544553018951225009, 5.66037360169780218180545218815, 6.64173360325397727413371338916, 7.74597652412341814988104030203, 8.971694411553695824216425530445, 10.18069604097718888816905520926, 11.34043325297840783623196539298

Graph of the ZZ-function along the critical line