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Properties

Label 2-39e2-1.1-c3-0-23
Degree 22
Conductor 15211521
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s − 9·5-s − 15·7-s − 21·8-s − 27·10-s − 48·11-s − 45·14-s − 71·16-s − 45·17-s − 6·19-s − 9·20-s − 144·22-s + 162·23-s − 44·25-s − 15·28-s + 144·29-s − 264·31-s − 45·32-s − 135·34-s + 135·35-s − 303·37-s − 18·38-s + 189·40-s − 192·41-s + 97·43-s − 48·44-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.804·5-s − 0.809·7-s − 0.928·8-s − 0.853·10-s − 1.31·11-s − 0.859·14-s − 1.10·16-s − 0.642·17-s − 0.0724·19-s − 0.100·20-s − 1.39·22-s + 1.46·23-s − 0.351·25-s − 0.101·28-s + 0.922·29-s − 1.52·31-s − 0.248·32-s − 0.680·34-s + 0.651·35-s − 1.34·37-s − 0.0768·38-s + 0.747·40-s − 0.731·41-s + 0.344·43-s − 0.164·44-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: 11
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: 00
Selberg data: (2, 1521, ( :3/2), 1)(2,\ 1521,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0189322551.018932255
L(12)L(\frac12) ≈\approx 1.0189322551.018932255
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+15T+p3T2 1 + 15 T + p^{3} T^{2}
11 1+48T+p3T2 1 + 48 T + p^{3} T^{2}
17 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
19 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
23 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
29 1−144T+p3T2 1 - 144 T + p^{3} T^{2}
31 1+264T+p3T2 1 + 264 T + p^{3} T^{2}
37 1+303T+p3T2 1 + 303 T + p^{3} T^{2}
41 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
43 1−97T+p3T2 1 - 97 T + p^{3} T^{2}
47 1−111T+p3T2 1 - 111 T + p^{3} T^{2}
53 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
59 1−522T+p3T2 1 - 522 T + p^{3} T^{2}
61 1−376T+p3T2 1 - 376 T + p^{3} T^{2}
67 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
71 1−357T+p3T2 1 - 357 T + p^{3} T^{2}
73 1−1098T+p3T2 1 - 1098 T + p^{3} T^{2}
79 1+830T+p3T2 1 + 830 T + p^{3} T^{2}
83 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
89 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
97 1−852T+p3T2 1 - 852 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

−8.976942760629447238606902445799, −8.358419432408291463467998485874, −7.25173457937375322870501830821, −6.64425485580142140233175617013, −5.49947688925844799235584599437, −4.99110435950193780490023977481, −3.94948274356930410424839514114, −3.29056901060642994485092905087, −2.41951943457579296897232703824, −0.39243248543384265385656516099, 0.39243248543384265385656516099, 2.41951943457579296897232703824, 3.29056901060642994485092905087, 3.94948274356930410424839514114, 4.99110435950193780490023977481, 5.49947688925844799235584599437, 6.64425485580142140233175617013, 7.25173457937375322870501830821, 8.358419432408291463467998485874, 8.976942760629447238606902445799

Graph of the ZZ-function along the critical line