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Properties

Label 2-162-1.1-c3-0-9
Degree 22
Conductor 162162
Sign −1-1
Analytic cond. 9.558309.55830
Root an. cond. 3.091653.09165
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 9·5-s − 31·7-s − 8·8-s − 18·10-s + 15·11-s − 37·13-s + 62·14-s + 16·16-s + 42·17-s − 28·19-s + 36·20-s − 30·22-s − 195·23-s − 44·25-s + 74·26-s − 124·28-s − 111·29-s − 205·31-s − 32·32-s − 84·34-s − 279·35-s − 166·37-s + 56·38-s − 72·40-s + 261·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.804·5-s − 1.67·7-s − 0.353·8-s − 0.569·10-s + 0.411·11-s − 0.789·13-s + 1.18·14-s + 1/4·16-s + 0.599·17-s − 0.338·19-s + 0.402·20-s − 0.290·22-s − 1.76·23-s − 0.351·25-s + 0.558·26-s − 0.836·28-s − 0.710·29-s − 1.18·31-s − 0.176·32-s − 0.423·34-s − 1.34·35-s − 0.737·37-s + 0.239·38-s − 0.284·40-s + 0.994·41-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −1-1
Analytic conductor: 9.558309.55830
Root analytic conductor: 3.091653.09165
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 162, ( :3/2), −1)(2,\ 162,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad2 1+pT 1 + p T
3 1 1
good5 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
7 1+31T+p3T2 1 + 31 T + p^{3} T^{2}
11 1−15T+p3T2 1 - 15 T + p^{3} T^{2}
13 1+37T+p3T2 1 + 37 T + p^{3} T^{2}
17 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
19 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
23 1+195T+p3T2 1 + 195 T + p^{3} T^{2}
29 1+111T+p3T2 1 + 111 T + p^{3} T^{2}
31 1+205T+p3T2 1 + 205 T + p^{3} T^{2}
37 1+166T+p3T2 1 + 166 T + p^{3} T^{2}
41 1−261T+p3T2 1 - 261 T + p^{3} T^{2}
43 1+pT+p3T2 1 + p T + p^{3} T^{2}
47 1+177T+p3T2 1 + 177 T + p^{3} T^{2}
53 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
59 1+159T+p3T2 1 + 159 T + p^{3} T^{2}
61 1−191T+p3T2 1 - 191 T + p^{3} T^{2}
67 1+421T+p3T2 1 + 421 T + p^{3} T^{2}
71 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
73 1−182T+p3T2 1 - 182 T + p^{3} T^{2}
79 1−1133T+p3T2 1 - 1133 T + p^{3} T^{2}
83 1−1083T+p3T2 1 - 1083 T + p^{3} T^{2}
89 1−1050T+p3T2 1 - 1050 T + p^{3} T^{2}
97 1+901T+p3T2 1 + 901 T + p^{3} 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

−12.00233694210455583132269709395, −10.49084250587472579635532929567, −9.708434387655531672536319104145, −9.235013556988423397630700181999, −7.68497038230648195921060986768, −6.52996595255696752320889082464, −5.73320443153005755121658124085, −3.60114648545183038779602919539, −2.10822466650392394975477454917, 0, 2.10822466650392394975477454917, 3.60114648545183038779602919539, 5.73320443153005755121658124085, 6.52996595255696752320889082464, 7.68497038230648195921060986768, 9.235013556988423397630700181999, 9.708434387655531672536319104145, 10.49084250587472579635532929567, 12.00233694210455583132269709395

Graph of the ZZ-function along the critical line