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Properties

Label 2-1872-1.1-c1-0-3
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  − 2.82·5-s − 0.828·7-s − 0.828·11-s + 13-s − 4·17-s − 0.828·19-s + 4·23-s + 3.00·25-s − 4·29-s + 10.4·31-s + 2.34·35-s + 2·37-s − 1.17·41-s + 5.65·43-s + 6.48·47-s − 6.31·49-s − 2.34·53-s + 2.34·55-s + 0.828·59-s + 9.31·61-s − 2.82·65-s + 0.828·67-s + 14.4·71-s + 6·73-s + 0.686·77-s + 4·79-s − 8.82·83-s + ⋯
L(s)  = 1  − 1.26·5-s − 0.313·7-s − 0.249·11-s + 0.277·13-s − 0.970·17-s − 0.190·19-s + 0.834·23-s + 0.600·25-s − 0.742·29-s + 1.88·31-s + 0.396·35-s + 0.328·37-s − 0.182·41-s + 0.862·43-s + 0.945·47-s − 0.901·49-s − 0.321·53-s + 0.315·55-s + 0.107·59-s + 1.19·61-s − 0.350·65-s + 0.101·67-s + 1.71·71-s + 0.702·73-s + 0.0782·77-s + 0.450·79-s − 0.969·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 1)(2,\ 1872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0680276901.068027690
L(12)L(\frac12) ≈\approx 1.0680276901.068027690
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
3 1 1
13 1−T 1 - T
good5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1+0.828T+7T2 1 + 0.828T + 7T^{2}
11 1+0.828T+11T2 1 + 0.828T + 11T^{2}
17 1+4T+17T2 1 + 4T + 17T^{2}
19 1+0.828T+19T2 1 + 0.828T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1−10.4T+31T2 1 - 10.4T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+1.17T+41T2 1 + 1.17T + 41T^{2}
43 1−5.65T+43T2 1 - 5.65T + 43T^{2}
47 1−6.48T+47T2 1 - 6.48T + 47T^{2}
53 1+2.34T+53T2 1 + 2.34T + 53T^{2}
59 1−0.828T+59T2 1 - 0.828T + 59T^{2}
61 1−9.31T+61T2 1 - 9.31T + 61T^{2}
67 1−0.828T+67T2 1 - 0.828T + 67T^{2}
71 1−14.4T+71T2 1 - 14.4T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+8.82T+83T2 1 + 8.82T + 83T^{2}
89 1−4.48T+89T2 1 - 4.48T + 89T^{2}
97 1−17.3T+97T2 1 - 17.3T + 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

−9.103908612675923452079127297892, −8.378029854677077491711979960021, −7.73011920376894555226821416921, −6.91482545291643159379085703303, −6.19315682071802761847884512389, −4.99862145026304318003221416791, −4.22702497616863086091706436193, −3.43449583228082154987019949962, −2.39409239714770356407675075129, −0.68307113866169886014610049280, 0.68307113866169886014610049280, 2.39409239714770356407675075129, 3.43449583228082154987019949962, 4.22702497616863086091706436193, 4.99862145026304318003221416791, 6.19315682071802761847884512389, 6.91482545291643159379085703303, 7.73011920376894555226821416921, 8.378029854677077491711979960021, 9.103908612675923452079127297892

Graph of the ZZ-function along the critical line