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Properties

Label 2-232050-1.1-c1-0-108
Degree 22
Conductor 232050232050
Sign 11
Analytic cond. 1852.921852.92
Root an. cond. 43.045643.0456
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 − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 4·11-s − 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 4·19-s − 21-s + 4·22-s + 4·23-s − 24-s − 26-s − 27-s + 28-s − 2·29-s + 8·31-s + 32-s − 4·33-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s + 0.834·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s − 0.171·34-s + ⋯

Functional equation

Λ(s)=(232050s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(232050s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 232050232050    =    2⋅3⋅52⋅7⋅13⋅172 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 1852.921852.92
Root analytic conductor: 43.045643.0456
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 232050, ( :1/2), 1)(2,\ 232050,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.9863094825.986309482
L(12)L(\frac12) ≈\approx 5.9863094825.986309482
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)
bad2 1−T 1 - T
3 1+T 1 + T
5 1 1
7 1−T 1 - T
13 1+T 1 + T
17 1+T 1 + T
good11 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−12.93644801926993, −12.41867443137779, −11.85588096359894, −11.69686320608495, −11.11874057615837, −10.94359122692321, −10.09236011645028, −9.750826349963517, −9.290931745757348, −8.696369960313880, −8.145359904032775, −7.503332289257483, −7.181704406226293, −6.551381354135759, −6.265972824955992, −5.640234827141397, −5.194561397690396, −4.646640457146668, −4.213554530877636, −3.764914652328716, −3.025481282162288, −2.511272650964124, −1.811014997634654, −1.023487934263096, −0.7536124537809807, 0.7536124537809807, 1.023487934263096, 1.811014997634654, 2.511272650964124, 3.025481282162288, 3.764914652328716, 4.213554530877636, 4.646640457146668, 5.194561397690396, 5.640234827141397, 6.265972824955992, 6.551381354135759, 7.181704406226293, 7.503332289257483, 8.145359904032775, 8.696369960313880, 9.290931745757348, 9.750826349963517, 10.09236011645028, 10.94359122692321, 11.11874057615837, 11.69686320608495, 11.85588096359894, 12.41867443137779, 12.93644801926993

Graph of the ZZ-function along the critical line