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Properties

Label 2-16245-1.1-c1-0-13
Degree 22
Conductor 1624516245
Sign 11
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 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s − 4·7-s − 3·11-s − 2·13-s + 4·16-s − 6·17-s − 2·20-s + 25-s + 8·28-s − 3·29-s + 7·31-s − 4·35-s − 8·37-s − 6·41-s − 4·43-s + 6·44-s − 6·47-s + 9·49-s + 4·52-s − 6·53-s − 3·55-s − 15·59-s + 5·61-s − 8·64-s − 2·65-s − 2·67-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 1.51·7-s − 0.904·11-s − 0.554·13-s + 16-s − 1.45·17-s − 0.447·20-s + 1/5·25-s + 1.51·28-s − 0.557·29-s + 1.25·31-s − 0.676·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.875·47-s + 9/7·49-s + 0.554·52-s − 0.824·53-s − 0.404·55-s − 1.95·59-s + 0.640·61-s − 64-s − 0.248·65-s − 0.244·67-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: 11
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: 22
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+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+3T+pT2 1 + 3 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+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+15T+pT2 1 + 15 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1+5T+pT2 1 + 5 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−16.56632611896340, −15.84489021353139, −15.44199610293729, −14.95234591070546, −13.96900344912967, −13.69556895497439, −13.20530000526345, −12.71244449233728, −12.37509453170336, −11.47117783613607, −10.66837217729375, −10.08270882424419, −9.783519736466002, −9.190506016905916, −8.620370611125498, −8.052672286828466, −7.145535631846718, −6.592271646650536, −6.022000586231617, −5.209001842139968, −4.760552971710474, −3.959278525319110, −3.144560563804680, −2.645225921828843, −1.573020251661425, 0, 0, 1.573020251661425, 2.645225921828843, 3.144560563804680, 3.959278525319110, 4.760552971710474, 5.209001842139968, 6.022000586231617, 6.592271646650536, 7.145535631846718, 8.052672286828466, 8.620370611125498, 9.190506016905916, 9.783519736466002, 10.08270882424419, 10.66837217729375, 11.47117783613607, 12.37509453170336, 12.71244449233728, 13.20530000526345, 13.69556895497439, 13.96900344912967, 14.95234591070546, 15.44199610293729, 15.84489021353139, 16.56632611896340

Graph of the ZZ-function along the critical line