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Properties

Label 2-105-1.1-c9-0-19
Degree 22
Conductor 105105
Sign 11
Analytic cond. 54.078754.0787
Root an. cond. 7.353827.35382
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 41.7·2-s − 81·3-s + 1.23e3·4-s − 625·5-s − 3.38e3·6-s + 2.40e3·7-s + 3.01e4·8-s + 6.56e3·9-s − 2.61e4·10-s − 1.90e4·11-s − 9.98e4·12-s + 1.72e5·13-s + 1.00e5·14-s + 5.06e4·15-s + 6.26e5·16-s − 2.53e5·17-s + 2.74e5·18-s + 4.04e4·19-s − 7.70e5·20-s − 1.94e5·21-s − 7.96e5·22-s + 1.52e6·23-s − 2.43e6·24-s + 3.90e5·25-s + 7.18e6·26-s − 5.31e5·27-s + 2.95e6·28-s + ⋯
L(s)  = 1  + 1.84·2-s − 0.577·3-s + 2.40·4-s − 0.447·5-s − 1.06·6-s + 0.377·7-s + 2.59·8-s + 0.333·9-s − 0.825·10-s − 0.392·11-s − 1.39·12-s + 1.67·13-s + 0.697·14-s + 0.258·15-s + 2.38·16-s − 0.736·17-s + 0.615·18-s + 0.0712·19-s − 1.07·20-s − 0.218·21-s − 0.724·22-s + 1.13·23-s − 1.50·24-s + 0.200·25-s + 3.08·26-s − 0.192·27-s + 0.909·28-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 105105    =    3⋅5⋅73 \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 54.078754.0787
Root analytic conductor: 7.353827.35382
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 105, ( :9/2), 1)(2,\ 105,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 6.1251763706.125176370
L(12)L(\frac12) ≈\approx 6.1251763706.125176370
L(112)L(\frac{11}{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+81T 1 + 81T
5 1+625T 1 + 625T
7 1−2.40e3T 1 - 2.40e3T
good2 1−41.7T+512T2 1 - 41.7T + 512T^{2}
11 1+1.90e4T+2.35e9T2 1 + 1.90e4T + 2.35e9T^{2}
13 1−1.72e5T+1.06e10T2 1 - 1.72e5T + 1.06e10T^{2}
17 1+2.53e5T+1.18e11T2 1 + 2.53e5T + 1.18e11T^{2}
19 1−4.04e4T+3.22e11T2 1 - 4.04e4T + 3.22e11T^{2}
23 1−1.52e6T+1.80e12T2 1 - 1.52e6T + 1.80e12T^{2}
29 1−4.68e6T+1.45e13T2 1 - 4.68e6T + 1.45e13T^{2}
31 1−6.33e6T+2.64e13T2 1 - 6.33e6T + 2.64e13T^{2}
37 1−8.76e6T+1.29e14T2 1 - 8.76e6T + 1.29e14T^{2}
41 1+2.45e7T+3.27e14T2 1 + 2.45e7T + 3.27e14T^{2}
43 1−2.34e7T+5.02e14T2 1 - 2.34e7T + 5.02e14T^{2}
47 1+1.96e7T+1.11e15T2 1 + 1.96e7T + 1.11e15T^{2}
53 1−8.41e7T+3.29e15T2 1 - 8.41e7T + 3.29e15T^{2}
59 1+7.51e7T+8.66e15T2 1 + 7.51e7T + 8.66e15T^{2}
61 1−5.29e7T+1.16e16T2 1 - 5.29e7T + 1.16e16T^{2}
67 1−1.37e8T+2.72e16T2 1 - 1.37e8T + 2.72e16T^{2}
71 1+3.28e8T+4.58e16T2 1 + 3.28e8T + 4.58e16T^{2}
73 1+2.17e8T+5.88e16T2 1 + 2.17e8T + 5.88e16T^{2}
79 1−2.26e8T+1.19e17T2 1 - 2.26e8T + 1.19e17T^{2}
83 1+2.45e8T+1.86e17T2 1 + 2.45e8T + 1.86e17T^{2}
89 1−1.80e8T+3.50e17T2 1 - 1.80e8T + 3.50e17T^{2}
97 1+3.85e8T+7.60e17T2 1 + 3.85e8T + 7.60e17T^{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

−12.04904719233437338036022007707, −11.27734275187878922148482597356, −10.60148964297346978706610069797, −8.393849210492640090775587799354, −6.95186532724292807329950040811, −6.07556134509474513744922988271, −4.94353969155180190866730386050, −4.06881969567614597666058745167, −2.81707418831635136005104752918, −1.17551097781652903446139097814, 1.17551097781652903446139097814, 2.81707418831635136005104752918, 4.06881969567614597666058745167, 4.94353969155180190866730386050, 6.07556134509474513744922988271, 6.95186532724292807329950040811, 8.393849210492640090775587799354, 10.60148964297346978706610069797, 11.27734275187878922148482597356, 12.04904719233437338036022007707

Graph of the ZZ-function along the critical line