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Properties

Label 2-16245-1.1-c1-0-8
Degree 22
Conductor 1624516245
Sign −1-1
Analytic cond. 129.716129.716
Root an. cond. 11.389311.3893
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s + 4·11-s − 2·13-s − 2·14-s − 16-s − 6·17-s − 20-s − 4·22-s − 6·23-s + 25-s + 2·26-s − 2·28-s − 4·31-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 3·40-s + 8·41-s + 6·43-s − 4·44-s + 6·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.223·20-s − 0.852·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.718·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.474·40-s + 1.24·41-s + 0.914·43-s − 0.603·44-s + 0.884·46-s + ⋯

Functional equation

Λ(s)=(16245s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(16245s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1624516245    =    32⋅5⋅1923^{2} \cdot 5 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 129.716129.716
Root analytic conductor: 11.389311.3893
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 16245, ( :1/2), −1)(2,\ 16245,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
5 1−T 1 - T
19 1 1
good2 1+T+pT2 1 + T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+18T+pT2 1 + 18 T + p 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

−16.26273937756855, −15.91727906104058, −14.88760424800857, −14.57938440149819, −13.94184778502602, −13.72267683213131, −12.70194159591985, −12.54615258919076, −11.46095993136365, −11.16924453010695, −10.58888832245443, −9.677299850500546, −9.476642633847634, −8.933325272809587, −8.244521579974750, −7.792224544737773, −7.044489712575237, −6.396132570885751, −5.694102927777490, −4.872375169060376, −4.286782733654280, −3.875108661727217, −2.508016437010309, −1.849992287269097, −1.117190259165496, 0, 1.117190259165496, 1.849992287269097, 2.508016437010309, 3.875108661727217, 4.286782733654280, 4.872375169060376, 5.694102927777490, 6.396132570885751, 7.044489712575237, 7.792224544737773, 8.244521579974750, 8.933325272809587, 9.476642633847634, 9.677299850500546, 10.58888832245443, 11.16924453010695, 11.46095993136365, 12.54615258919076, 12.70194159591985, 13.72267683213131, 13.94184778502602, 14.57938440149819, 14.88760424800857, 15.91727906104058, 16.26273937756855

Graph of the ZZ-function along the critical line