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Properties

Label 2-476-1.1-c1-0-2
Degree 22
Conductor 476476
Sign 11
Analytic cond. 3.800873.80087
Root an. cond. 1.949581.94958
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  + 1.30·3-s + 2.30·5-s − 7-s − 1.30·9-s + 4·11-s + 2.60·13-s + 3·15-s − 17-s + 1.39·19-s − 1.30·21-s + 4·23-s + 0.302·25-s − 5.60·27-s − 5.21·29-s + 3.69·31-s + 5.21·33-s − 2.30·35-s − 11.8·37-s + 3.39·39-s + 6.51·41-s − 0.697·43-s − 3.00·45-s + 4.60·47-s + 49-s − 1.30·51-s + 4.30·53-s + 9.21·55-s + ⋯
L(s)  = 1  + 0.752·3-s + 1.02·5-s − 0.377·7-s − 0.434·9-s + 1.20·11-s + 0.722·13-s + 0.774·15-s − 0.242·17-s + 0.319·19-s − 0.284·21-s + 0.834·23-s + 0.0605·25-s − 1.07·27-s − 0.967·29-s + 0.664·31-s + 0.907·33-s − 0.389·35-s − 1.94·37-s + 0.543·39-s + 1.01·41-s − 0.106·43-s − 0.447·45-s + 0.671·47-s + 0.142·49-s − 0.182·51-s + 0.591·53-s + 1.24·55-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 3.800873.80087
Root analytic conductor: 1.949581.94958
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 476, ( :1/2), 1)(2,\ 476,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0692449302.069244930
L(12)L(\frac12) ≈\approx 2.0692449302.069244930
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)
bad2 1 1
7 1+T 1 + T
17 1+T 1 + T
good3 1−1.30T+3T2 1 - 1.30T + 3T^{2}
5 1−2.30T+5T2 1 - 2.30T + 5T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−2.60T+13T2 1 - 2.60T + 13T^{2}
19 1−1.39T+19T2 1 - 1.39T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+5.21T+29T2 1 + 5.21T + 29T^{2}
31 1−3.69T+31T2 1 - 3.69T + 31T^{2}
37 1+11.8T+37T2 1 + 11.8T + 37T^{2}
41 1−6.51T+41T2 1 - 6.51T + 41T^{2}
43 1+0.697T+43T2 1 + 0.697T + 43T^{2}
47 1−4.60T+47T2 1 - 4.60T + 47T^{2}
53 1−4.30T+53T2 1 - 4.30T + 53T^{2}
59 1+8T+59T2 1 + 8T + 59T^{2}
61 1+2.51T+61T2 1 + 2.51T + 61T^{2}
67 1−6.30T+67T2 1 - 6.30T + 67T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1+10.5T+73T2 1 + 10.5T + 73T^{2}
79 1−4.60T+79T2 1 - 4.60T + 79T^{2}
83 1+11.2T+83T2 1 + 11.2T + 83T^{2}
89 1+13.8T+89T2 1 + 13.8T + 89T^{2}
97 1+9.69T+97T2 1 + 9.69T + 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

−10.95314083862301695146980952542, −9.883127184923822987309935775602, −9.084510549202141237522329932422, −8.691166707003106484924568493522, −7.31021489287142883396217234915, −6.30036227906475194920740174462, −5.53163241973076577201627516367, −3.96049769465594348872907084961, −2.92304355581998684891020819472, −1.61734174675707997492615665344, 1.61734174675707997492615665344, 2.92304355581998684891020819472, 3.96049769465594348872907084961, 5.53163241973076577201627516367, 6.30036227906475194920740174462, 7.31021489287142883396217234915, 8.691166707003106484924568493522, 9.084510549202141237522329932422, 9.883127184923822987309935775602, 10.95314083862301695146980952542

Graph of the ZZ-function along the critical line