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Properties

Label 2-39e2-1.1-c3-0-124
Degree 22
Conductor 15211521
Sign −1-1
Analytic cond. 89.741989.7419
Root an. cond. 9.473229.47322
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  − 3·2-s + 4-s + 9·5-s + 2·7-s + 21·8-s − 27·10-s − 30·11-s − 6·14-s − 71·16-s + 111·17-s − 46·19-s + 9·20-s + 90·22-s + 6·23-s − 44·25-s + 2·28-s + 105·29-s − 100·31-s + 45·32-s − 333·34-s + 18·35-s + 17·37-s + 138·38-s + 189·40-s + 231·41-s − 514·43-s − 30·44-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s + 0.804·5-s + 0.107·7-s + 0.928·8-s − 0.853·10-s − 0.822·11-s − 0.114·14-s − 1.10·16-s + 1.58·17-s − 0.555·19-s + 0.100·20-s + 0.872·22-s + 0.0543·23-s − 0.351·25-s + 0.0134·28-s + 0.672·29-s − 0.579·31-s + 0.248·32-s − 1.67·34-s + 0.0869·35-s + 0.0755·37-s + 0.589·38-s + 0.747·40-s + 0.879·41-s − 1.82·43-s − 0.102·44-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 89.741989.7419
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1521, ( :3/2), −1)(2,\ 1521,\ (\ :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)
bad3 1 1
13 1 1
good2 1+3T+p3T2 1 + 3 T + p^{3} T^{2}
5 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
7 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
17 1−111T+p3T2 1 - 111 T + p^{3} T^{2}
19 1+46T+p3T2 1 + 46 T + p^{3} T^{2}
23 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
29 1−105T+p3T2 1 - 105 T + p^{3} T^{2}
31 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
37 1−17T+p3T2 1 - 17 T + p^{3} T^{2}
41 1−231T+p3T2 1 - 231 T + p^{3} T^{2}
43 1+514T+p3T2 1 + 514 T + p^{3} T^{2}
47 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
53 1+639T+p3T2 1 + 639 T + p^{3} T^{2}
59 1+600T+p3T2 1 + 600 T + p^{3} T^{2}
61 1−233T+p3T2 1 - 233 T + p^{3} T^{2}
67 1−926T+p3T2 1 - 926 T + p^{3} T^{2}
71 1−930T+p3T2 1 - 930 T + p^{3} T^{2}
73 1+253T+p3T2 1 + 253 T + p^{3} T^{2}
79 1+1324T+p3T2 1 + 1324 T + p^{3} T^{2}
83 1+810T+p3T2 1 + 810 T + p^{3} T^{2}
89 1+498T+p3T2 1 + 498 T + p^{3} T^{2}
97 1−14pT+p3T2 1 - 14 p 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

−8.682972386261179174557038859205, −8.038263293687297801116545838310, −7.41643940349391247560926189412, −6.32589935209881622990059876288, −5.42873358453323535741815580340, −4.66475584683334183300667883871, −3.33846548891653449816670222871, −2.12091531081045564354298857925, −1.22140671015271970789027629264, 0, 1.22140671015271970789027629264, 2.12091531081045564354298857925, 3.33846548891653449816670222871, 4.66475584683334183300667883871, 5.42873358453323535741815580340, 6.32589935209881622990059876288, 7.41643940349391247560926189412, 8.038263293687297801116545838310, 8.682972386261179174557038859205

Graph of the ZZ-function along the critical line