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Properties

Label 2-1859-1.1-c3-0-313
Degree 22
Conductor 18591859
Sign −1-1
Analytic cond. 109.684109.684
Root an. cond. 10.473010.4730
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.231·2-s + 8.68·3-s − 7.94·4-s − 0.297·5-s + 2.01·6-s − 7.88·7-s − 3.69·8-s + 48.4·9-s − 0.0690·10-s − 11·11-s − 69.0·12-s − 1.82·14-s − 2.58·15-s + 62.7·16-s − 15.3·17-s + 11.2·18-s − 120.·19-s + 2.36·20-s − 68.4·21-s − 2.55·22-s + 156.·23-s − 32.1·24-s − 124.·25-s + 186.·27-s + 62.6·28-s + 217.·29-s − 0.599·30-s + ⋯
L(s)  = 1  + 0.0819·2-s + 1.67·3-s − 0.993·4-s − 0.0266·5-s + 0.137·6-s − 0.425·7-s − 0.163·8-s + 1.79·9-s − 0.00218·10-s − 0.301·11-s − 1.66·12-s − 0.0349·14-s − 0.0445·15-s + 0.979·16-s − 0.219·17-s + 0.147·18-s − 1.45·19-s + 0.0264·20-s − 0.711·21-s − 0.0247·22-s + 1.41·23-s − 0.273·24-s − 0.999·25-s + 1.33·27-s + 0.422·28-s + 1.39·29-s − 0.00364·30-s + ⋯

Functional equation

Λ(s)=(1859s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1859s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18591859    =    11⋅13211 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 109.684109.684
Root analytic conductor: 10.473010.4730
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1859, ( :3/2), −1)(2,\ 1859,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad11 1+11T 1 + 11T
13 1 1
good2 1−0.231T+8T2 1 - 0.231T + 8T^{2}
3 1−8.68T+27T2 1 - 8.68T + 27T^{2}
5 1+0.297T+125T2 1 + 0.297T + 125T^{2}
7 1+7.88T+343T2 1 + 7.88T + 343T^{2}
17 1+15.3T+4.91e3T2 1 + 15.3T + 4.91e3T^{2}
19 1+120.T+6.85e3T2 1 + 120.T + 6.85e3T^{2}
23 1−156.T+1.21e4T2 1 - 156.T + 1.21e4T^{2}
29 1−217.T+2.43e4T2 1 - 217.T + 2.43e4T^{2}
31 1−209.T+2.97e4T2 1 - 209.T + 2.97e4T^{2}
37 1−182.T+5.06e4T2 1 - 182.T + 5.06e4T^{2}
41 1+286.T+6.89e4T2 1 + 286.T + 6.89e4T^{2}
43 1+419.T+7.95e4T2 1 + 419.T + 7.95e4T^{2}
47 1+196.T+1.03e5T2 1 + 196.T + 1.03e5T^{2}
53 1+175.T+1.48e5T2 1 + 175.T + 1.48e5T^{2}
59 1+185.T+2.05e5T2 1 + 185.T + 2.05e5T^{2}
61 1−140.T+2.26e5T2 1 - 140.T + 2.26e5T^{2}
67 1−484.T+3.00e5T2 1 - 484.T + 3.00e5T^{2}
71 1+833.T+3.57e5T2 1 + 833.T + 3.57e5T^{2}
73 1+1.15e3T+3.89e5T2 1 + 1.15e3T + 3.89e5T^{2}
79 1+147.T+4.93e5T2 1 + 147.T + 4.93e5T^{2}
83 1+737.T+5.71e5T2 1 + 737.T + 5.71e5T^{2}
89 1+1.19e3T+7.04e5T2 1 + 1.19e3T + 7.04e5T^{2}
97 1−422.T+9.12e5T2 1 - 422.T + 9.12e5T^{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

−8.355751413449869094815796636329, −8.234270799205112007115804222399, −7.05863634901222320135973396015, −6.20980723854287796996810894946, −4.83524715878985797704519536412, −4.27903677446040746004482084535, −3.29016176690569524472338542910, −2.70735796087711096205712051995, −1.46007588523744801658049136796, 0, 1.46007588523744801658049136796, 2.70735796087711096205712051995, 3.29016176690569524472338542910, 4.27903677446040746004482084535, 4.83524715878985797704519536412, 6.20980723854287796996810894946, 7.05863634901222320135973396015, 8.234270799205112007115804222399, 8.355751413449869094815796636329

Graph of the ZZ-function along the critical line