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Properties

Label 2-1859-1.1-c3-0-346
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  + 2.30·2-s + 4.54·3-s − 2.69·4-s + 4.16·5-s + 10.4·6-s + 17.3·7-s − 24.6·8-s − 6.31·9-s + 9.59·10-s − 11·11-s − 12.2·12-s + 39.9·14-s + 18.9·15-s − 35.1·16-s + 1.39·17-s − 14.5·18-s − 58.9·19-s − 11.2·20-s + 78.9·21-s − 25.3·22-s + 46.6·23-s − 112.·24-s − 107.·25-s − 151.·27-s − 46.8·28-s + 175.·29-s + 43.6·30-s + ⋯
L(s)  = 1  + 0.814·2-s + 0.875·3-s − 0.337·4-s + 0.372·5-s + 0.712·6-s + 0.937·7-s − 1.08·8-s − 0.233·9-s + 0.303·10-s − 0.301·11-s − 0.295·12-s + 0.763·14-s + 0.326·15-s − 0.548·16-s + 0.0198·17-s − 0.190·18-s − 0.711·19-s − 0.125·20-s + 0.820·21-s − 0.245·22-s + 0.422·23-s − 0.952·24-s − 0.861·25-s − 1.07·27-s − 0.316·28-s + 1.12·29-s + 0.265·30-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 1−2.30T+8T2 1 - 2.30T + 8T^{2}
3 1−4.54T+27T2 1 - 4.54T + 27T^{2}
5 1−4.16T+125T2 1 - 4.16T + 125T^{2}
7 1−17.3T+343T2 1 - 17.3T + 343T^{2}
17 1−1.39T+4.91e3T2 1 - 1.39T + 4.91e3T^{2}
19 1+58.9T+6.85e3T2 1 + 58.9T + 6.85e3T^{2}
23 1−46.6T+1.21e4T2 1 - 46.6T + 1.21e4T^{2}
29 1−175.T+2.43e4T2 1 - 175.T + 2.43e4T^{2}
31 1−33.3T+2.97e4T2 1 - 33.3T + 2.97e4T^{2}
37 1+376.T+5.06e4T2 1 + 376.T + 5.06e4T^{2}
41 1−270.T+6.89e4T2 1 - 270.T + 6.89e4T^{2}
43 1+125.T+7.95e4T2 1 + 125.T + 7.95e4T^{2}
47 1+562.T+1.03e5T2 1 + 562.T + 1.03e5T^{2}
53 1+356.T+1.48e5T2 1 + 356.T + 1.48e5T^{2}
59 1+138.T+2.05e5T2 1 + 138.T + 2.05e5T^{2}
61 1−358.T+2.26e5T2 1 - 358.T + 2.26e5T^{2}
67 1+291.T+3.00e5T2 1 + 291.T + 3.00e5T^{2}
71 1+90.2T+3.57e5T2 1 + 90.2T + 3.57e5T^{2}
73 1+366.T+3.89e5T2 1 + 366.T + 3.89e5T^{2}
79 1−270.T+4.93e5T2 1 - 270.T + 4.93e5T^{2}
83 1−746.T+5.71e5T2 1 - 746.T + 5.71e5T^{2}
89 1+1.48e3T+7.04e5T2 1 + 1.48e3T + 7.04e5T^{2}
97 1+869.T+9.12e5T2 1 + 869.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

−8.328233533810927181728073915769, −8.083569678810670858936521014432, −6.78559992190276303349938837998, −5.85457073418438657485443515007, −5.09279537417072453553555738147, −4.39738544846530001953675589684, −3.42479103593178067151506518103, −2.63122548914633838450505543583, −1.64216054188870969321051858664, 0, 1.64216054188870969321051858664, 2.63122548914633838450505543583, 3.42479103593178067151506518103, 4.39738544846530001953675589684, 5.09279537417072453553555738147, 5.85457073418438657485443515007, 6.78559992190276303349938837998, 8.083569678810670858936521014432, 8.328233533810927181728073915769

Graph of the ZZ-function along the critical line