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Properties

Label 2-4304-1.1-c1-0-35
Degree 22
Conductor 43044304
Sign 11
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
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  − 1.65·3-s + 0.725·5-s − 1.29·7-s − 0.269·9-s + 3.71·11-s + 2.30·13-s − 1.19·15-s + 6.32·17-s − 0.0212·19-s + 2.14·21-s + 1.30·23-s − 4.47·25-s + 5.40·27-s − 2.06·29-s + 5.82·31-s − 6.14·33-s − 0.942·35-s − 7.25·37-s − 3.81·39-s + 8.77·41-s − 4.52·43-s − 0.195·45-s − 3.85·47-s − 5.31·49-s − 10.4·51-s − 6.86·53-s + 2.69·55-s + ⋯
L(s)  = 1  − 0.954·3-s + 0.324·5-s − 0.491·7-s − 0.0896·9-s + 1.12·11-s + 0.640·13-s − 0.309·15-s + 1.53·17-s − 0.00487·19-s + 0.468·21-s + 0.272·23-s − 0.894·25-s + 1.03·27-s − 0.383·29-s + 1.04·31-s − 1.06·33-s − 0.159·35-s − 1.19·37-s − 0.610·39-s + 1.37·41-s − 0.689·43-s − 0.0290·45-s − 0.562·47-s − 0.758·49-s − 1.46·51-s − 0.943·53-s + 0.363·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: 11
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4304, ( :1/2), 1)(2,\ 4304,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4120966331.412096633
L(12)L(\frac12) ≈\approx 1.4120966331.412096633
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
269 1+T 1 + T
good3 1+1.65T+3T2 1 + 1.65T + 3T^{2}
5 1−0.725T+5T2 1 - 0.725T + 5T^{2}
7 1+1.29T+7T2 1 + 1.29T + 7T^{2}
11 1−3.71T+11T2 1 - 3.71T + 11T^{2}
13 1−2.30T+13T2 1 - 2.30T + 13T^{2}
17 1−6.32T+17T2 1 - 6.32T + 17T^{2}
19 1+0.0212T+19T2 1 + 0.0212T + 19T^{2}
23 1−1.30T+23T2 1 - 1.30T + 23T^{2}
29 1+2.06T+29T2 1 + 2.06T + 29T^{2}
31 1−5.82T+31T2 1 - 5.82T + 31T^{2}
37 1+7.25T+37T2 1 + 7.25T + 37T^{2}
41 1−8.77T+41T2 1 - 8.77T + 41T^{2}
43 1+4.52T+43T2 1 + 4.52T + 43T^{2}
47 1+3.85T+47T2 1 + 3.85T + 47T^{2}
53 1+6.86T+53T2 1 + 6.86T + 53T^{2}
59 1+7.84T+59T2 1 + 7.84T + 59T^{2}
61 1−3.84T+61T2 1 - 3.84T + 61T^{2}
67 1−9.31T+67T2 1 - 9.31T + 67T^{2}
71 1−0.643T+71T2 1 - 0.643T + 71T^{2}
73 1+2.97T+73T2 1 + 2.97T + 73T^{2}
79 1+0.418T+79T2 1 + 0.418T + 79T^{2}
83 1−15.6T+83T2 1 - 15.6T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−5.07T+97T2 1 - 5.07T + 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

−8.372228525960342914380569649162, −7.61375687970555994711177676759, −6.55100430544147304714094861982, −6.25032696596805641788441595539, −5.57040258060339934528516231920, −4.82061949452151046088171028374, −3.73766046773862869153350459832, −3.11518416840174933991605960703, −1.69330086616727328574300114513, −0.73536141609062732824911942677, 0.73536141609062732824911942677, 1.69330086616727328574300114513, 3.11518416840174933991605960703, 3.73766046773862869153350459832, 4.82061949452151046088171028374, 5.57040258060339934528516231920, 6.25032696596805641788441595539, 6.55100430544147304714094861982, 7.61375687970555994711177676759, 8.372228525960342914380569649162

Graph of the ZZ-function along the critical line