↑

Properties

Label 2-285-1.1-c9-0-25
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  − 32.1·2-s − 81·3-s + 520.·4-s + 625·5-s + 2.60e3·6-s + 7.29e3·7-s − 264.·8-s + 6.56e3·9-s − 2.00e4·10-s − 7.92e4·11-s − 4.21e4·12-s + 1.48e5·13-s − 2.34e5·14-s − 5.06e4·15-s − 2.57e5·16-s − 4.03e5·17-s − 2.10e5·18-s − 1.30e5·19-s + 3.25e5·20-s − 5.90e5·21-s + 2.54e6·22-s + 1.87e6·23-s + 2.14e4·24-s + 3.90e5·25-s − 4.76e6·26-s − 5.31e5·27-s + 3.79e6·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1.01·4-s + 0.447·5-s + 0.819·6-s + 1.14·7-s − 0.0228·8-s + 0.333·9-s − 0.634·10-s − 1.63·11-s − 0.586·12-s + 1.43·13-s − 1.62·14-s − 0.258·15-s − 0.983·16-s − 1.17·17-s − 0.473·18-s − 0.229·19-s + 0.454·20-s − 0.662·21-s + 2.31·22-s + 1.39·23-s + 0.0131·24-s + 0.200·25-s − 2.04·26-s − 0.192·27-s + 1.16·28-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 0.93838662580.9383866258
L(12)L(\frac12) ≈\approx 0.93838662580.9383866258
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+32.1T+512T2 1 + 32.1T + 512T^{2}
7 1−7.29e3T+4.03e7T2 1 - 7.29e3T + 4.03e7T^{2}
11 1+7.92e4T+2.35e9T2 1 + 7.92e4T + 2.35e9T^{2}
13 1−1.48e5T+1.06e10T2 1 - 1.48e5T + 1.06e10T^{2}
17 1+4.03e5T+1.18e11T2 1 + 4.03e5T + 1.18e11T^{2}
23 1−1.87e6T+1.80e12T2 1 - 1.87e6T + 1.80e12T^{2}
29 1+4.55e6T+1.45e13T2 1 + 4.55e6T + 1.45e13T^{2}
31 1−7.02e6T+2.64e13T2 1 - 7.02e6T + 2.64e13T^{2}
37 1−6.13e6T+1.29e14T2 1 - 6.13e6T + 1.29e14T^{2}
41 1−8.76e6T+3.27e14T2 1 - 8.76e6T + 3.27e14T^{2}
43 1−1.06e7T+5.02e14T2 1 - 1.06e7T + 5.02e14T^{2}
47 1+2.81e7T+1.11e15T2 1 + 2.81e7T + 1.11e15T^{2}
53 1−7.72e7T+3.29e15T2 1 - 7.72e7T + 3.29e15T^{2}
59 1+6.20e7T+8.66e15T2 1 + 6.20e7T + 8.66e15T^{2}
61 1−5.26e7T+1.16e16T2 1 - 5.26e7T + 1.16e16T^{2}
67 1−2.09e8T+2.72e16T2 1 - 2.09e8T + 2.72e16T^{2}
71 1+3.12e8T+4.58e16T2 1 + 3.12e8T + 4.58e16T^{2}
73 1+2.93e8T+5.88e16T2 1 + 2.93e8T + 5.88e16T^{2}
79 1−1.63e8T+1.19e17T2 1 - 1.63e8T + 1.19e17T^{2}
83 1−7.23e8T+1.86e17T2 1 - 7.23e8T + 1.86e17T^{2}
89 1−3.62e8T+3.50e17T2 1 - 3.62e8T + 3.50e17T^{2}
97 1+3.40e8T+7.60e17T2 1 + 3.40e8T + 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

−10.42225837441992889778293262528, −9.196708868473622003201282487413, −8.404929579584721252218413667200, −7.68504254328550733887589756105, −6.58214406745222901100507305420, −5.39325193519507914284706887359, −4.46555903399599638431357520036, −2.48769041241462520657387353538, −1.49133106620254065270928848559, −0.59207277838887156509392874591, 0.59207277838887156509392874591, 1.49133106620254065270928848559, 2.48769041241462520657387353538, 4.46555903399599638431357520036, 5.39325193519507914284706887359, 6.58214406745222901100507305420, 7.68504254328550733887589756105, 8.404929579584721252218413667200, 9.196708868473622003201282487413, 10.42225837441992889778293262528

Graph of the ZZ-function along the critical line