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Properties

Label 2-2368-1.1-c1-0-51
Degree 22
Conductor 23682368
Sign 11
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
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  + 3.30·3-s + 2.30·5-s + 2.60·7-s + 7.90·9-s − 2.30·11-s − 1.30·13-s + 7.60·15-s − 6·17-s + 2·19-s + 8.60·21-s − 3.90·23-s + 0.302·25-s + 16.2·27-s + 3.90·29-s + 0.302·31-s − 7.60·33-s + 6·35-s − 37-s − 4.30·39-s + 9.90·41-s + 0.605·43-s + 18.2·45-s − 4.60·47-s − 0.211·49-s − 19.8·51-s + 6·53-s − 5.30·55-s + ⋯
L(s)  = 1  + 1.90·3-s + 1.02·5-s + 0.984·7-s + 2.63·9-s − 0.694·11-s − 0.361·13-s + 1.96·15-s − 1.45·17-s + 0.458·19-s + 1.87·21-s − 0.814·23-s + 0.0605·25-s + 3.11·27-s + 0.725·29-s + 0.0543·31-s − 1.32·33-s + 1.01·35-s − 0.164·37-s − 0.688·39-s + 1.54·41-s + 0.0923·43-s + 2.71·45-s − 0.671·47-s − 0.0301·49-s − 2.77·51-s + 0.824·53-s − 0.715·55-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: 11
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2368, ( :1/2), 1)(2,\ 2368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.6144677024.614467702
L(12)L(\frac12) ≈\approx 4.6144677024.614467702
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
37 1+T 1 + T
good3 1−3.30T+3T2 1 - 3.30T + 3T^{2}
5 1−2.30T+5T2 1 - 2.30T + 5T^{2}
7 1−2.60T+7T2 1 - 2.60T + 7T^{2}
11 1+2.30T+11T2 1 + 2.30T + 11T^{2}
13 1+1.30T+13T2 1 + 1.30T + 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+3.90T+23T2 1 + 3.90T + 23T^{2}
29 1−3.90T+29T2 1 - 3.90T + 29T^{2}
31 1−0.302T+31T2 1 - 0.302T + 31T^{2}
41 1−9.90T+41T2 1 - 9.90T + 41T^{2}
43 1−0.605T+43T2 1 - 0.605T + 43T^{2}
47 1+4.60T+47T2 1 + 4.60T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−10.6T+59T2 1 - 10.6T + 59T^{2}
61 1+7.51T+61T2 1 + 7.51T + 61T^{2}
67 1+3.51T+67T2 1 + 3.51T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1+9.11T+79T2 1 + 9.11T + 79T^{2}
83 1−2.78T+83T2 1 - 2.78T + 83T^{2}
89 1+9.21T+89T2 1 + 9.21T + 89T^{2}
97 1+16.4T+97T2 1 + 16.4T + 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

−8.855820573373444028373712414413, −8.336611144699361093517820113497, −7.64429795463427911698833468025, −6.95825527448039212767536842635, −5.84019526948162706012289293039, −4.74679733400694768212541219378, −4.14541829434390951103105683313, −2.83802902324237826115557032885, −2.27347514723706435329817268742, −1.55299958308258776591813379532, 1.55299958308258776591813379532, 2.27347514723706435329817268742, 2.83802902324237826115557032885, 4.14541829434390951103105683313, 4.74679733400694768212541219378, 5.84019526948162706012289293039, 6.95825527448039212767536842635, 7.64429795463427911698833468025, 8.336611144699361093517820113497, 8.855820573373444028373712414413

Graph of the ZZ-function along the critical line