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Properties

Label 2-1840-1.1-c1-0-0
Degree 22
Conductor 18401840
Sign 11
Analytic cond. 14.692414.6924
Root an. cond. 3.833073.83307
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  − 1.93·3-s − 5-s − 2.38·7-s + 0.735·9-s − 5.33·11-s − 4.53·13-s + 1.93·15-s + 1.81·17-s − 7.00·19-s + 4.60·21-s + 23-s + 25-s + 4.37·27-s − 0.118·29-s + 0.884·31-s + 10.3·33-s + 2.38·35-s + 7.51·37-s + 8.77·39-s − 1.45·41-s − 0.735·45-s − 10.4·47-s − 1.32·49-s − 3.50·51-s − 9.42·53-s + 5.33·55-s + 13.5·57-s + ⋯
L(s)  = 1  − 1.11·3-s − 0.447·5-s − 0.900·7-s + 0.245·9-s − 1.60·11-s − 1.25·13-s + 0.499·15-s + 0.440·17-s − 1.60·19-s + 1.00·21-s + 0.208·23-s + 0.200·25-s + 0.842·27-s − 0.0219·29-s + 0.158·31-s + 1.79·33-s + 0.402·35-s + 1.23·37-s + 1.40·39-s − 0.226·41-s − 0.109·45-s − 1.52·47-s − 0.189·49-s − 0.491·51-s − 1.29·53-s + 0.719·55-s + 1.79·57-s + ⋯

Functional equation

Λ(s)=(1840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18401840    =    24⋅5⋅232^{4} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 14.692414.6924
Root analytic conductor: 3.833073.83307
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1840, ( :1/2), 1)(2,\ 1840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.24788136560.2478813656
L(12)L(\frac12) ≈\approx 0.24788136560.2478813656
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
5 1+T 1 + T
23 1−T 1 - T
good3 1+1.93T+3T2 1 + 1.93T + 3T^{2}
7 1+2.38T+7T2 1 + 2.38T + 7T^{2}
11 1+5.33T+11T2 1 + 5.33T + 11T^{2}
13 1+4.53T+13T2 1 + 4.53T + 13T^{2}
17 1−1.81T+17T2 1 - 1.81T + 17T^{2}
19 1+7.00T+19T2 1 + 7.00T + 19T^{2}
29 1+0.118T+29T2 1 + 0.118T + 29T^{2}
31 1−0.884T+31T2 1 - 0.884T + 31T^{2}
37 1−7.51T+37T2 1 - 7.51T + 37T^{2}
41 1+1.45T+41T2 1 + 1.45T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+10.4T+47T2 1 + 10.4T + 47T^{2}
53 1+9.42T+53T2 1 + 9.42T + 53T^{2}
59 1+7.79T+59T2 1 + 7.79T + 59T^{2}
61 1+2.80T+61T2 1 + 2.80T + 61T^{2}
67 1−3.11T+67T2 1 - 3.11T + 67T^{2}
71 1−13.5T+71T2 1 - 13.5T + 71T^{2}
73 1−12.4T+73T2 1 - 12.4T + 73T^{2}
79 1+6.80T+79T2 1 + 6.80T + 79T^{2}
83 1−13.5T+83T2 1 - 13.5T + 83T^{2}
89 1−2.89T+89T2 1 - 2.89T + 89T^{2}
97 1+1.97T+97T2 1 + 1.97T + 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

−9.491403231926375014812525892123, −8.257065013165311266685521909733, −7.69066969321607573581739248795, −6.66916328489374909934527829284, −6.12595048408386719338227348832, −5.10963416088376515193759381939, −4.64863388003871234103167066276, −3.25875493881566695097223488837, −2.37603137320533751218488741797, −0.32957100908932285842405515070, 0.32957100908932285842405515070, 2.37603137320533751218488741797, 3.25875493881566695097223488837, 4.64863388003871234103167066276, 5.10963416088376515193759381939, 6.12595048408386719338227348832, 6.66916328489374909934527829284, 7.69066969321607573581739248795, 8.257065013165311266685521909733, 9.491403231926375014812525892123

Graph of the ZZ-function along the critical line