↑

Properties

Label 2-285-1.1-c9-0-8
Degree 22
Conductor 285285
Sign 11
Analytic cond. 146.785146.785
Root an. cond. 12.115412.1154
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 17.4·2-s − 81·3-s − 206.·4-s + 625·5-s + 1.41e3·6-s − 3.26e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 1.09e4·10-s − 1.63e4·11-s + 1.67e4·12-s − 1.68e5·13-s + 5.71e4·14-s − 5.06e4·15-s − 1.13e5·16-s − 3.97e5·17-s − 1.14e5·18-s − 1.30e5·19-s − 1.28e5·20-s + 2.64e5·21-s + 2.85e5·22-s + 1.77e6·23-s − 1.01e6·24-s + 3.90e5·25-s + 2.95e6·26-s − 5.31e5·27-s + 6.74e5·28-s + ⋯
L(s)  = 1  − 0.772·2-s − 0.577·3-s − 0.403·4-s + 0.447·5-s + 0.446·6-s − 0.514·7-s + 1.08·8-s + 0.333·9-s − 0.345·10-s − 0.335·11-s + 0.232·12-s − 1.63·13-s + 0.397·14-s − 0.258·15-s − 0.434·16-s − 1.15·17-s − 0.257·18-s − 0.229·19-s − 0.180·20-s + 0.297·21-s + 0.259·22-s + 1.32·23-s − 0.625·24-s + 0.200·25-s + 1.26·26-s − 0.192·27-s + 0.207·28-s + ⋯

Functional equation

Λ(s)=(285s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(285s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 285285    =    3⋅5⋅193 \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 146.785146.785
Root analytic conductor: 12.115412.1154
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 285, ( :9/2), 1)(2,\ 285,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.26178747230.2617874723
L(12)L(\frac12) ≈\approx 0.26178747230.2617874723
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
19 1+1.30e5T 1 + 1.30e5T
good2 1+17.4T+512T2 1 + 17.4T + 512T^{2}
7 1+3.26e3T+4.03e7T2 1 + 3.26e3T + 4.03e7T^{2}
11 1+1.63e4T+2.35e9T2 1 + 1.63e4T + 2.35e9T^{2}
13 1+1.68e5T+1.06e10T2 1 + 1.68e5T + 1.06e10T^{2}
17 1+3.97e5T+1.18e11T2 1 + 3.97e5T + 1.18e11T^{2}
23 1−1.77e6T+1.80e12T2 1 - 1.77e6T + 1.80e12T^{2}
29 1−4.10e6T+1.45e13T2 1 - 4.10e6T + 1.45e13T^{2}
31 1−7.49e5T+2.64e13T2 1 - 7.49e5T + 2.64e13T^{2}
37 1+1.40e7T+1.29e14T2 1 + 1.40e7T + 1.29e14T^{2}
41 1−7.51e6T+3.27e14T2 1 - 7.51e6T + 3.27e14T^{2}
43 1+3.50e7T+5.02e14T2 1 + 3.50e7T + 5.02e14T^{2}
47 1+1.98e7T+1.11e15T2 1 + 1.98e7T + 1.11e15T^{2}
53 1−6.86e7T+3.29e15T2 1 - 6.86e7T + 3.29e15T^{2}
59 1+1.18e8T+8.66e15T2 1 + 1.18e8T + 8.66e15T^{2}
61 1+1.76e8T+1.16e16T2 1 + 1.76e8T + 1.16e16T^{2}
67 1+7.82e7T+2.72e16T2 1 + 7.82e7T + 2.72e16T^{2}
71 1−1.99e8T+4.58e16T2 1 - 1.99e8T + 4.58e16T^{2}
73 1+2.70e8T+5.88e16T2 1 + 2.70e8T + 5.88e16T^{2}
79 1+5.51e8T+1.19e17T2 1 + 5.51e8T + 1.19e17T^{2}
83 1+2.14e7T+1.86e17T2 1 + 2.14e7T + 1.86e17T^{2}
89 1−1.76e8T+3.50e17T2 1 - 1.76e8T + 3.50e17T^{2}
97 1+1.58e8T+7.60e17T2 1 + 1.58e8T + 7.60e17T^{2}
show more
show less
   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

−10.14470189346395547465108766987, −9.411199191645607592367966533217, −8.562996871454934725499972110786, −7.31916188568729187321102473693, −6.57788422435322637249609859724, −5.13972122795359851457945703855, −4.56112284399463518974203783124, −2.84268216709228098473908879856, −1.58677567998265041905642075103, −0.26770276494045140076290568463, 0.26770276494045140076290568463, 1.58677567998265041905642075103, 2.84268216709228098473908879856, 4.56112284399463518974203783124, 5.13972122795359851457945703855, 6.57788422435322637249609859724, 7.31916188568729187321102473693, 8.562996871454934725499972110786, 9.411199191645607592367966533217, 10.14470189346395547465108766987

Graph of the ZZ-function along the critical line