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Properties

Label 2-275-1.1-c1-0-7
Degree 22
Conductor 275275
Sign 11
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s − 1.30·3-s + 3.30·4-s − 3·6-s + 4.30·7-s + 3.00·8-s − 1.30·9-s − 11-s − 4.30·12-s + 5·13-s + 9.90·14-s + 0.302·16-s − 3.90·17-s − 3.00·18-s − 19-s − 5.60·21-s − 2.30·22-s − 3.69·23-s − 3.90·24-s + 11.5·26-s + 5.60·27-s + 14.2·28-s − 9.90·29-s − 4.21·31-s − 5.30·32-s + 1.30·33-s − 9·34-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.752·3-s + 1.65·4-s − 1.22·6-s + 1.62·7-s + 1.06·8-s − 0.434·9-s − 0.301·11-s − 1.24·12-s + 1.38·13-s + 2.64·14-s + 0.0756·16-s − 0.947·17-s − 0.707·18-s − 0.229·19-s − 1.22·21-s − 0.490·22-s − 0.770·23-s − 0.797·24-s + 2.25·26-s + 1.07·27-s + 2.68·28-s − 1.83·29-s − 0.756·31-s − 0.937·32-s + 0.226·33-s − 1.54·34-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 11
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), 1)(2,\ 275,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5438351462.543835146
L(12)L(\frac12) ≈\approx 2.5438351462.543835146
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)
bad5 1 1
11 1+T 1 + T
good2 1−2.30T+2T2 1 - 2.30T + 2T^{2}
3 1+1.30T+3T2 1 + 1.30T + 3T^{2}
7 1−4.30T+7T2 1 - 4.30T + 7T^{2}
13 1−5T+13T2 1 - 5T + 13T^{2}
17 1+3.90T+17T2 1 + 3.90T + 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+3.69T+23T2 1 + 3.69T + 23T^{2}
29 1+9.90T+29T2 1 + 9.90T + 29T^{2}
31 1+4.21T+31T2 1 + 4.21T + 31T^{2}
37 1−9.60T+37T2 1 - 9.60T + 37T^{2}
41 1−1.60T+41T2 1 - 1.60T + 41T^{2}
43 1+7.21T+43T2 1 + 7.21T + 43T^{2}
47 1+3T+47T2 1 + 3T + 47T^{2}
53 1−2.30T+53T2 1 - 2.30T + 53T^{2}
59 1−0.211T+59T2 1 - 0.211T + 59T^{2}
61 1−2.90T+61T2 1 - 2.90T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−4.60T+71T2 1 - 4.60T + 71T^{2}
73 1−2.90T+73T2 1 - 2.90T + 73T^{2}
79 1+0.0916T+79T2 1 + 0.0916T + 79T^{2}
83 1−14.5T+83T2 1 - 14.5T + 83T^{2}
89 1−5.30T+89T2 1 - 5.30T + 89T^{2}
97 1−11.6T+97T2 1 - 11.6T + 97T^{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

−11.75264289167998397713824240051, −11.24933123400379278318604389692, −10.83460400463024479003249245451, −8.843291587354579709770843927933, −7.78167051977930937994868351532, −6.36993791897891590419911293865, −5.62137348147238836438259296365, −4.80175039695978728299343645479, −3.79894828953799276588994147306, −2.04290806640930588735681357089, 2.04290806640930588735681357089, 3.79894828953799276588994147306, 4.80175039695978728299343645479, 5.62137348147238836438259296365, 6.36993791897891590419911293865, 7.78167051977930937994868351532, 8.843291587354579709770843927933, 10.83460400463024479003249245451, 11.24933123400379278318604389692, 11.75264289167998397713824240051

Graph of the ZZ-function along the critical line