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Properties

Label 2-304-1.1-c1-0-2
Degree 22
Conductor 304304
Sign 11
Analytic cond. 2.427452.42745
Root an. cond. 1.558021.55802
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  − 0.786·3-s + 3.29·5-s + 2.08·7-s − 2.38·9-s − 1.29·11-s + 1.21·13-s − 2.59·15-s + 4.08·17-s + 19-s − 1.63·21-s + 8.95·23-s + 5.87·25-s + 4.23·27-s − 9.38·29-s + 1.02·33-s + 6.87·35-s − 2·37-s − 0.954·39-s + 3.57·41-s − 7.72·43-s − 7.85·45-s − 9.46·47-s − 2.65·49-s − 3.21·51-s − 11.9·53-s − 4.27·55-s − 0.786·57-s + ⋯
L(s)  = 1  − 0.454·3-s + 1.47·5-s + 0.787·7-s − 0.793·9-s − 0.391·11-s + 0.336·13-s − 0.669·15-s + 0.990·17-s + 0.229·19-s − 0.357·21-s + 1.86·23-s + 1.17·25-s + 0.814·27-s − 1.74·29-s + 0.177·33-s + 1.16·35-s − 0.328·37-s − 0.152·39-s + 0.558·41-s − 1.17·43-s − 1.17·45-s − 1.38·47-s − 0.379·49-s − 0.449·51-s − 1.64·53-s − 0.576·55-s − 0.104·57-s + ⋯

Functional equation

Λ(s)=(304s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(304s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 304304    =    24⋅192^{4} \cdot 19
Sign: 11
Analytic conductor: 2.427452.42745
Root analytic conductor: 1.558021.55802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 304, ( :1/2), 1)(2,\ 304,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4586683331.458668333
L(12)L(\frac12) ≈\approx 1.4586683331.458668333
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
19 1−T 1 - T
good3 1+0.786T+3T2 1 + 0.786T + 3T^{2}
5 1−3.29T+5T2 1 - 3.29T + 5T^{2}
7 1−2.08T+7T2 1 - 2.08T + 7T^{2}
11 1+1.29T+11T2 1 + 1.29T + 11T^{2}
13 1−1.21T+13T2 1 - 1.21T + 13T^{2}
17 1−4.08T+17T2 1 - 4.08T + 17T^{2}
23 1−8.95T+23T2 1 - 8.95T + 23T^{2}
29 1+9.38T+29T2 1 + 9.38T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1−3.57T+41T2 1 - 3.57T + 41T^{2}
43 1+7.72T+43T2 1 + 7.72T + 43T^{2}
47 1+9.46T+47T2 1 + 9.46T + 47T^{2}
53 1+11.9T+53T2 1 + 11.9T + 53T^{2}
59 1−7.21T+59T2 1 - 7.21T + 59T^{2}
61 1−4.87T+61T2 1 - 4.87T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−9.02T+71T2 1 - 9.02T + 71T^{2}
73 1−5.65T+73T2 1 - 5.65T + 73T^{2}
79 1+9.57T+79T2 1 + 9.57T + 79T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 1−11.0T+89T2 1 - 11.0T + 89T^{2}
97 1+8.59T+97T2 1 + 8.59T + 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

−11.46041065289235852729454214063, −10.91552549995755363480443255575, −9.869133929594604236173049500670, −9.015355166181452972281499650357, −7.946775557002075432635587888010, −6.63093008896007893457608729078, −5.52399931717913474337243460244, −5.13009513375226831578289402134, −3.05986132748281541374303515005, −1.56727195113053216494868612301, 1.56727195113053216494868612301, 3.05986132748281541374303515005, 5.13009513375226831578289402134, 5.52399931717913474337243460244, 6.63093008896007893457608729078, 7.946775557002075432635587888010, 9.015355166181452972281499650357, 9.869133929594604236173049500670, 10.91552549995755363480443255575, 11.46041065289235852729454214063

Graph of the ZZ-function along the critical line