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Properties

Label 2-18e2-1.1-c3-0-0
Degree 22
Conductor 324324
Sign 11
Analytic cond. 19.116619.1166
Root an. cond. 4.372254.37225
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  − 21.0·5-s − 31.4·7-s − 36.6·11-s + 56.4·13-s + 35.8·17-s + 83.4·19-s + 69.5·23-s + 318.·25-s − 81.7·29-s − 72.9·31-s + 663.·35-s − 25.4·37-s − 399.·41-s − 83.4·43-s − 311.·47-s + 648.·49-s + 4.09·53-s + 772.·55-s − 352.·59-s + 3.54·61-s − 1.19e3·65-s + 492.·67-s + 154.·71-s + 305·73-s + 1.15e3·77-s + 671.·79-s + 1.29e3·83-s + ⋯
L(s)  = 1  − 1.88·5-s − 1.70·7-s − 1.00·11-s + 1.20·13-s + 0.510·17-s + 1.00·19-s + 0.630·23-s + 2.55·25-s − 0.523·29-s − 0.422·31-s + 3.20·35-s − 0.113·37-s − 1.52·41-s − 0.295·43-s − 0.967·47-s + 1.89·49-s + 0.0106·53-s + 1.89·55-s − 0.777·59-s + 0.00743·61-s − 2.27·65-s + 0.897·67-s + 0.258·71-s + 0.489·73-s + 1.70·77-s + 0.956·79-s + 1.71·83-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: 11
Analytic conductor: 19.116619.1166
Root analytic conductor: 4.372254.37225
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 324, ( :3/2), 1)(2,\ 324,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.69876548680.6987654868
L(12)L(\frac12) ≈\approx 0.69876548680.6987654868
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 1
3 1 1
good5 1+21.0T+125T2 1 + 21.0T + 125T^{2}
7 1+31.4T+343T2 1 + 31.4T + 343T^{2}
11 1+36.6T+1.33e3T2 1 + 36.6T + 1.33e3T^{2}
13 1−56.4T+2.19e3T2 1 - 56.4T + 2.19e3T^{2}
17 1−35.8T+4.91e3T2 1 - 35.8T + 4.91e3T^{2}
19 1−83.4T+6.85e3T2 1 - 83.4T + 6.85e3T^{2}
23 1−69.5T+1.21e4T2 1 - 69.5T + 1.21e4T^{2}
29 1+81.7T+2.43e4T2 1 + 81.7T + 2.43e4T^{2}
31 1+72.9T+2.97e4T2 1 + 72.9T + 2.97e4T^{2}
37 1+25.4T+5.06e4T2 1 + 25.4T + 5.06e4T^{2}
41 1+399.T+6.89e4T2 1 + 399.T + 6.89e4T^{2}
43 1+83.4T+7.95e4T2 1 + 83.4T + 7.95e4T^{2}
47 1+311.T+1.03e5T2 1 + 311.T + 1.03e5T^{2}
53 1−4.09T+1.48e5T2 1 - 4.09T + 1.48e5T^{2}
59 1+352.T+2.05e5T2 1 + 352.T + 2.05e5T^{2}
61 1−3.54T+2.26e5T2 1 - 3.54T + 2.26e5T^{2}
67 1−492.T+3.00e5T2 1 - 492.T + 3.00e5T^{2}
71 1−154.T+3.57e5T2 1 - 154.T + 3.57e5T^{2}
73 1−305T+3.89e5T2 1 - 305T + 3.89e5T^{2}
79 1−671.T+4.93e5T2 1 - 671.T + 4.93e5T^{2}
83 1−1.29e3T+5.71e5T2 1 - 1.29e3T + 5.71e5T^{2}
89 1−1.18e3T+7.04e5T2 1 - 1.18e3T + 7.04e5T^{2}
97 1−615.T+9.12e5T2 1 - 615.T + 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

−11.23074928489637685115718442322, −10.37750211206190403011451802029, −9.252258462132100365959047866578, −8.223208813124219231094464471305, −7.40966339375449554894075932387, −6.49045365419211824066673174331, −5.11964057969035803684936801117, −3.54263595492155741376837018996, −3.28041010413970998434169783772, −0.55116993514232491671914777466, 0.55116993514232491671914777466, 3.28041010413970998434169783772, 3.54263595492155741376837018996, 5.11964057969035803684936801117, 6.49045365419211824066673174331, 7.40966339375449554894075932387, 8.223208813124219231094464471305, 9.252258462132100365959047866578, 10.37750211206190403011451802029, 11.23074928489637685115718442322

Graph of the ZZ-function along the critical line