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Properties

Label 2-16245-1.1-c1-0-5
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 00

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 + 6·11-s + 2·14-s − 16-s + 6·17-s − 20-s − 6·22-s + 8·23-s + 25-s + 2·28-s + 4·29-s − 5·32-s − 6·34-s − 2·35-s − 4·37-s + 3·40-s − 2·43-s − 6·44-s − 8·46-s + 8·47-s − 3·49-s − 50-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 + 1.80·11-s + 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.223·20-s − 1.27·22-s + 1.66·23-s + 1/5·25-s + 0.377·28-s + 0.742·29-s − 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.657·37-s + 0.474·40-s − 0.304·43-s − 0.904·44-s − 1.17·46-s + 1.16·47-s − 3/7·49-s − 0.141·50-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: 00
Selberg data: (2, 16245, ( :1/2), 1)(2,\ 16245,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8698665931.869866593
L(12)L(\frac12) ≈\approx 1.8698665931.869866593
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 1−6T+pT2 1 - 6 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−12T+pT2 1 - 12 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

−16.16399751103832, −15.47257990165858, −14.64243959211847, −14.26818697844889, −13.87854692735768, −13.11272662172290, −12.65752680449208, −12.07802927678784, −11.41333201617904, −10.71048389731151, −10.02432593217566, −9.682951225770028, −9.161805288550961, −8.706585790936350, −8.090658814035643, −7.129098887693482, −6.837314628943155, −6.093636523863323, −5.304373770366630, −4.733967184017791, −3.678871812133920, −3.472829506572583, −2.274984101139950, −1.162684181867679, −0.8389474356698935, 0.8389474356698935, 1.162684181867679, 2.274984101139950, 3.472829506572583, 3.678871812133920, 4.733967184017791, 5.304373770366630, 6.093636523863323, 6.837314628943155, 7.129098887693482, 8.090658814035643, 8.706585790936350, 9.161805288550961, 9.682951225770028, 10.02432593217566, 10.71048389731151, 11.41333201617904, 12.07802927678784, 12.65752680449208, 13.11272662172290, 13.87854692735768, 14.26818697844889, 14.64243959211847, 15.47257990165858, 16.16399751103832

Graph of the ZZ-function along the critical line