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Properties

Label 2-18e2-1.1-c3-0-6
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  + 13.8·5-s + 30.7·7-s + 43.9·11-s − 12.2·13-s − 76.0·17-s − 44.1·19-s + 78.6·23-s + 66.6·25-s + 92.7·29-s − 143.·31-s + 425.·35-s − 32.4·37-s + 335.·41-s − 498.·43-s + 281.·47-s + 599.·49-s + 628.·53-s + 607.·55-s − 504.·59-s + 371.·61-s − 169.·65-s − 162.·67-s − 433.·71-s − 629.·73-s + 1.34e3·77-s + 172.·79-s + 174.·83-s + ⋯
L(s)  = 1  + 1.23·5-s + 1.65·7-s + 1.20·11-s − 0.261·13-s − 1.08·17-s − 0.533·19-s + 0.712·23-s + 0.533·25-s + 0.594·29-s − 0.828·31-s + 2.05·35-s − 0.144·37-s + 1.27·41-s − 1.76·43-s + 0.874·47-s + 1.74·49-s + 1.62·53-s + 1.49·55-s − 1.11·59-s + 0.780·61-s − 0.323·65-s − 0.296·67-s − 0.724·71-s − 1.00·73-s + 1.99·77-s + 0.245·79-s + 0.231·83-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 2.8649311752.864931175
L(12)L(\frac12) ≈\approx 2.8649311752.864931175
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 1−13.8T+125T2 1 - 13.8T + 125T^{2}
7 1−30.7T+343T2 1 - 30.7T + 343T^{2}
11 1−43.9T+1.33e3T2 1 - 43.9T + 1.33e3T^{2}
13 1+12.2T+2.19e3T2 1 + 12.2T + 2.19e3T^{2}
17 1+76.0T+4.91e3T2 1 + 76.0T + 4.91e3T^{2}
19 1+44.1T+6.85e3T2 1 + 44.1T + 6.85e3T^{2}
23 1−78.6T+1.21e4T2 1 - 78.6T + 1.21e4T^{2}
29 1−92.7T+2.43e4T2 1 - 92.7T + 2.43e4T^{2}
31 1+143.T+2.97e4T2 1 + 143.T + 2.97e4T^{2}
37 1+32.4T+5.06e4T2 1 + 32.4T + 5.06e4T^{2}
41 1−335.T+6.89e4T2 1 - 335.T + 6.89e4T^{2}
43 1+498.T+7.95e4T2 1 + 498.T + 7.95e4T^{2}
47 1−281.T+1.03e5T2 1 - 281.T + 1.03e5T^{2}
53 1−628.T+1.48e5T2 1 - 628.T + 1.48e5T^{2}
59 1+504.T+2.05e5T2 1 + 504.T + 2.05e5T^{2}
61 1−371.T+2.26e5T2 1 - 371.T + 2.26e5T^{2}
67 1+162.T+3.00e5T2 1 + 162.T + 3.00e5T^{2}
71 1+433.T+3.57e5T2 1 + 433.T + 3.57e5T^{2}
73 1+629.T+3.89e5T2 1 + 629.T + 3.89e5T^{2}
79 1−172.T+4.93e5T2 1 - 172.T + 4.93e5T^{2}
83 1−174.T+5.71e5T2 1 - 174.T + 5.71e5T^{2}
89 1+336.T+7.04e5T2 1 + 336.T + 7.04e5T^{2}
97 1−84.3T+9.12e5T2 1 - 84.3T + 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

−11.16502808497292810386443012774, −10.33631837722078395718682948532, −9.152036577930500540670138871157, −8.607281576638410956659068198216, −7.26355717778189446115713514171, −6.25017850298695457988834554870, −5.17176860769255246391722148474, −4.23545895023832252490074674592, −2.27233095316729874050757091284, −1.37462613473195221734714052449, 1.37462613473195221734714052449, 2.27233095316729874050757091284, 4.23545895023832252490074674592, 5.17176860769255246391722148474, 6.25017850298695457988834554870, 7.26355717778189446115713514171, 8.607281576638410956659068198216, 9.152036577930500540670138871157, 10.33631837722078395718682948532, 11.16502808497292810386443012774

Graph of the ZZ-function along the critical line