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Properties

Label 2-1859-1.1-c3-0-347
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  + 1.40·2-s + 8.50·3-s − 6.01·4-s + 10.6·5-s + 11.9·6-s − 26.4·7-s − 19.7·8-s + 45.3·9-s + 14.9·10-s − 11·11-s − 51.1·12-s − 37.2·14-s + 90.2·15-s + 20.2·16-s + 3.00·17-s + 63.8·18-s + 164.·19-s − 63.8·20-s − 224.·21-s − 15.5·22-s − 194.·23-s − 167.·24-s − 12.3·25-s + 155.·27-s + 158.·28-s − 11.1·29-s + 127.·30-s + ⋯
L(s)  = 1  + 0.498·2-s + 1.63·3-s − 0.751·4-s + 0.949·5-s + 0.815·6-s − 1.42·7-s − 0.872·8-s + 1.67·9-s + 0.473·10-s − 0.301·11-s − 1.23·12-s − 0.710·14-s + 1.55·15-s + 0.316·16-s + 0.0428·17-s + 0.836·18-s + 1.98·19-s − 0.713·20-s − 2.33·21-s − 0.150·22-s − 1.76·23-s − 1.42·24-s − 0.0991·25-s + 1.11·27-s + 1.07·28-s − 0.0713·29-s + 0.774·30-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 1−1.40T+8T2 1 - 1.40T + 8T^{2}
3 1−8.50T+27T2 1 - 8.50T + 27T^{2}
5 1−10.6T+125T2 1 - 10.6T + 125T^{2}
7 1+26.4T+343T2 1 + 26.4T + 343T^{2}
17 1−3.00T+4.91e3T2 1 - 3.00T + 4.91e3T^{2}
19 1−164.T+6.85e3T2 1 - 164.T + 6.85e3T^{2}
23 1+194.T+1.21e4T2 1 + 194.T + 1.21e4T^{2}
29 1+11.1T+2.43e4T2 1 + 11.1T + 2.43e4T^{2}
31 1+326.T+2.97e4T2 1 + 326.T + 2.97e4T^{2}
37 1−179.T+5.06e4T2 1 - 179.T + 5.06e4T^{2}
41 1+501.T+6.89e4T2 1 + 501.T + 6.89e4T^{2}
43 1+334.T+7.95e4T2 1 + 334.T + 7.95e4T^{2}
47 1−298.T+1.03e5T2 1 - 298.T + 1.03e5T^{2}
53 1+284.T+1.48e5T2 1 + 284.T + 1.48e5T^{2}
59 1−525.T+2.05e5T2 1 - 525.T + 2.05e5T^{2}
61 1+177.T+2.26e5T2 1 + 177.T + 2.26e5T^{2}
67 1−365.T+3.00e5T2 1 - 365.T + 3.00e5T^{2}
71 1+225.T+3.57e5T2 1 + 225.T + 3.57e5T^{2}
73 1−76.3T+3.89e5T2 1 - 76.3T + 3.89e5T^{2}
79 1+886.T+4.93e5T2 1 + 886.T + 4.93e5T^{2}
83 1+247.T+5.71e5T2 1 + 247.T + 5.71e5T^{2}
89 1+1.18e3T+7.04e5T2 1 + 1.18e3T + 7.04e5T^{2}
97 1+420.T+9.12e5T2 1 + 420.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

−8.651002046684958410845067827117, −7.83992969783945076593264881538, −6.96630196713096935950183403311, −5.90416201092009023700813766629, −5.31181456000849014019682860456, −3.93950505163050128253088690986, −3.41221842561522648766497640264, −2.73069589731527154703998628464, −1.65732842884487028907108437018, 0, 1.65732842884487028907108437018, 2.73069589731527154703998628464, 3.41221842561522648766497640264, 3.93950505163050128253088690986, 5.31181456000849014019682860456, 5.90416201092009023700813766629, 6.96630196713096935950183403311, 7.83992969783945076593264881538, 8.651002046684958410845067827117

Graph of the ZZ-function along the critical line