↑

Properties

Label 2-285-1.1-c9-0-84
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  + 42.1·2-s − 81·3-s + 1.26e3·4-s + 625·5-s − 3.41e3·6-s + 4.30e3·7-s + 3.17e4·8-s + 6.56e3·9-s + 2.63e4·10-s + 7.42e4·11-s − 1.02e5·12-s + 1.62e5·13-s + 1.81e5·14-s − 5.06e4·15-s + 6.91e5·16-s + 3.54e5·17-s + 2.76e5·18-s − 1.30e5·19-s + 7.91e5·20-s − 3.48e5·21-s + 3.12e6·22-s − 2.24e6·23-s − 2.57e6·24-s + 3.90e5·25-s + 6.86e6·26-s − 5.31e5·27-s + 5.45e6·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.47·4-s + 0.447·5-s − 1.07·6-s + 0.678·7-s + 2.74·8-s + 0.333·9-s + 0.833·10-s + 1.52·11-s − 1.42·12-s + 1.58·13-s + 1.26·14-s − 0.258·15-s + 2.63·16-s + 1.03·17-s + 0.621·18-s − 0.229·19-s + 1.10·20-s − 0.391·21-s + 2.84·22-s − 1.67·23-s − 1.58·24-s + 0.200·25-s + 2.94·26-s − 0.192·27-s + 1.67·28-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 9.9580466279.958046627
L(12)L(\frac12) ≈\approx 9.9580466279.958046627
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 1−42.1T+512T2 1 - 42.1T + 512T^{2}
7 1−4.30e3T+4.03e7T2 1 - 4.30e3T + 4.03e7T^{2}
11 1−7.42e4T+2.35e9T2 1 - 7.42e4T + 2.35e9T^{2}
13 1−1.62e5T+1.06e10T2 1 - 1.62e5T + 1.06e10T^{2}
17 1−3.54e5T+1.18e11T2 1 - 3.54e5T + 1.18e11T^{2}
23 1+2.24e6T+1.80e12T2 1 + 2.24e6T + 1.80e12T^{2}
29 1+6.17e6T+1.45e13T2 1 + 6.17e6T + 1.45e13T^{2}
31 1+7.22e6T+2.64e13T2 1 + 7.22e6T + 2.64e13T^{2}
37 1−1.44e7T+1.29e14T2 1 - 1.44e7T + 1.29e14T^{2}
41 1+2.14e7T+3.27e14T2 1 + 2.14e7T + 3.27e14T^{2}
43 1−3.51e7T+5.02e14T2 1 - 3.51e7T + 5.02e14T^{2}
47 1+3.01e6T+1.11e15T2 1 + 3.01e6T + 1.11e15T^{2}
53 1+5.03e6T+3.29e15T2 1 + 5.03e6T + 3.29e15T^{2}
59 1−3.96e7T+8.66e15T2 1 - 3.96e7T + 8.66e15T^{2}
61 1−4.80e7T+1.16e16T2 1 - 4.80e7T + 1.16e16T^{2}
67 1−6.24e7T+2.72e16T2 1 - 6.24e7T + 2.72e16T^{2}
71 1−1.78e8T+4.58e16T2 1 - 1.78e8T + 4.58e16T^{2}
73 1−3.85e8T+5.88e16T2 1 - 3.85e8T + 5.88e16T^{2}
79 1+2.10e8T+1.19e17T2 1 + 2.10e8T + 1.19e17T^{2}
83 1−4.38e8T+1.86e17T2 1 - 4.38e8T + 1.86e17T^{2}
89 1+5.81e8T+3.50e17T2 1 + 5.81e8T + 3.50e17T^{2}
97 1+1.26e9T+7.60e17T2 1 + 1.26e9T + 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

−10.91567974195468930260311833470, −9.521530801394288290753516188399, −7.982311126817638678027758762992, −6.76035681095565787096460150944, −5.95258028396022398203676837081, −5.45770427771996681674743981004, −4.06242053349414453138764260961, −3.69156201432432069791864999326, −1.94148939953761022316329903198, −1.27056266863966110286936949603, 1.27056266863966110286936949603, 1.94148939953761022316329903198, 3.69156201432432069791864999326, 4.06242053349414453138764260961, 5.45770427771996681674743981004, 5.95258028396022398203676837081, 6.76035681095565787096460150944, 7.982311126817638678027758762992, 9.521530801394288290753516188399, 10.91567974195468930260311833470

Graph of the ZZ-function along the critical line