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Properties

Label 2-6e4-1.1-c3-0-27
Degree 22
Conductor 12961296
Sign 11
Analytic cond. 76.466476.4664
Root an. cond. 8.744518.74451
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s + 31·7-s − 15·11-s − 37·13-s + 42·17-s + 28·19-s + 195·23-s − 44·25-s − 111·29-s + 205·31-s + 279·35-s − 166·37-s + 261·41-s + 43·43-s + 177·47-s + 618·49-s − 114·53-s − 135·55-s + 159·59-s + 191·61-s − 333·65-s + 421·67-s + 156·71-s + 182·73-s − 465·77-s − 1.13e3·79-s − 1.08e3·83-s + ⋯
L(s)  = 1  + 0.804·5-s + 1.67·7-s − 0.411·11-s − 0.789·13-s + 0.599·17-s + 0.338·19-s + 1.76·23-s − 0.351·25-s − 0.710·29-s + 1.18·31-s + 1.34·35-s − 0.737·37-s + 0.994·41-s + 0.152·43-s + 0.549·47-s + 1.80·49-s − 0.295·53-s − 0.330·55-s + 0.350·59-s + 0.400·61-s − 0.635·65-s + 0.767·67-s + 0.260·71-s + 0.291·73-s − 0.688·77-s − 1.61·79-s − 1.43·83-s + ⋯

Functional equation

Λ(s)=(1296s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1296s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12961296    =    24⋅342^{4} \cdot 3^{4}
Sign: 11
Analytic conductor: 76.466476.4664
Root analytic conductor: 8.744518.74451
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1296, ( :3/2), 1)(2,\ 1296,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2612290883.261229088
L(12)L(\frac12) ≈\approx 3.2612290883.261229088
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 1−9T+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

−9.320273173894260984714815690609, −8.424815599384315930766334221184, −7.67759974536129000334768677359, −6.98205811420524123855911819601, −5.63574592115388834166004270235, −5.19240098143359732203485962259, −4.35613785306754757519506359359, −2.87344381244706326024240919920, −1.94269867512205913631379968250, −0.968643922886922191920894929279, 0.968643922886922191920894929279, 1.94269867512205913631379968250, 2.87344381244706326024240919920, 4.35613785306754757519506359359, 5.19240098143359732203485962259, 5.63574592115388834166004270235, 6.98205811420524123855911819601, 7.67759974536129000334768677359, 8.424815599384315930766334221184, 9.320273173894260984714815690609

Graph of the ZZ-function along the critical line