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Properties

Label 2-285-1.1-c9-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.1·2-s − 81·3-s − 107.·4-s + 625·5-s − 1.62e3·6-s − 2.84e3·7-s − 1.24e4·8-s + 6.56e3·9-s + 1.25e4·10-s − 9.64e4·11-s + 8.72e3·12-s − 1.18e5·13-s − 5.71e4·14-s − 5.06e4·15-s − 1.95e5·16-s − 3.67e5·17-s + 1.31e5·18-s − 1.30e5·19-s − 6.73e4·20-s + 2.30e5·21-s − 1.94e6·22-s + 2.13e5·23-s + 1.00e6·24-s + 3.90e5·25-s − 2.37e6·26-s − 5.31e5·27-s + 3.05e5·28-s + ⋯
L(s)  = 1  + 0.888·2-s − 0.577·3-s − 0.210·4-s + 0.447·5-s − 0.513·6-s − 0.447·7-s − 1.07·8-s + 0.333·9-s + 0.397·10-s − 1.98·11-s + 0.121·12-s − 1.14·13-s − 0.397·14-s − 0.258·15-s − 0.745·16-s − 1.06·17-s + 0.296·18-s − 0.229·19-s − 0.0940·20-s + 0.258·21-s − 1.76·22-s + 0.158·23-s + 0.620·24-s + 0.200·25-s − 1.02·26-s − 0.192·27-s + 0.0940·28-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 0.40119270780.4011927078
L(12)L(\frac12) ≈\approx 0.40119270780.4011927078
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 1−20.1T+512T2 1 - 20.1T + 512T^{2}
7 1+2.84e3T+4.03e7T2 1 + 2.84e3T + 4.03e7T^{2}
11 1+9.64e4T+2.35e9T2 1 + 9.64e4T + 2.35e9T^{2}
13 1+1.18e5T+1.06e10T2 1 + 1.18e5T + 1.06e10T^{2}
17 1+3.67e5T+1.18e11T2 1 + 3.67e5T + 1.18e11T^{2}
23 1−2.13e5T+1.80e12T2 1 - 2.13e5T + 1.80e12T^{2}
29 1+4.06e5T+1.45e13T2 1 + 4.06e5T + 1.45e13T^{2}
31 1−2.17e6T+2.64e13T2 1 - 2.17e6T + 2.64e13T^{2}
37 1+9.81e6T+1.29e14T2 1 + 9.81e6T + 1.29e14T^{2}
41 1+1.41e7T+3.27e14T2 1 + 1.41e7T + 3.27e14T^{2}
43 1−3.97e7T+5.02e14T2 1 - 3.97e7T + 5.02e14T^{2}
47 1+5.46e7T+1.11e15T2 1 + 5.46e7T + 1.11e15T^{2}
53 1+8.10e7T+3.29e15T2 1 + 8.10e7T + 3.29e15T^{2}
59 1−2.82e7T+8.66e15T2 1 - 2.82e7T + 8.66e15T^{2}
61 1−1.30e8T+1.16e16T2 1 - 1.30e8T + 1.16e16T^{2}
67 1+8.18e7T+2.72e16T2 1 + 8.18e7T + 2.72e16T^{2}
71 1+2.87e8T+4.58e16T2 1 + 2.87e8T + 4.58e16T^{2}
73 1−2.53e8T+5.88e16T2 1 - 2.53e8T + 5.88e16T^{2}
79 1−1.21e8T+1.19e17T2 1 - 1.21e8T + 1.19e17T^{2}
83 1+3.80e8T+1.86e17T2 1 + 3.80e8T + 1.86e17T^{2}
89 1+7.94e8T+3.50e17T2 1 + 7.94e8T + 3.50e17T^{2}
97 1−9.23e8T+7.60e17T2 1 - 9.23e8T + 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

−10.26630825730414974054352591825, −9.538913030652291672985430100560, −8.315940218183122124595887993617, −7.06435525646744944848339279726, −6.03721121859098707001086294915, −5.12914210484004856600216039570, −4.61326483959738372900405736310, −3.09846983515463609816922146139, −2.22558988025326436626518903809, −0.23500203592126998992320428987, 0.23500203592126998992320428987, 2.22558988025326436626518903809, 3.09846983515463609816922146139, 4.61326483959738372900405736310, 5.12914210484004856600216039570, 6.03721121859098707001086294915, 7.06435525646744944848339279726, 8.315940218183122124595887993617, 9.538913030652291672985430100560, 10.26630825730414974054352591825

Graph of the ZZ-function along the critical line