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Properties

Label 2-1859-1.1-c3-0-271
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  − 4.48·2-s − 3.05·3-s + 12.0·4-s + 15.8·5-s + 13.6·6-s + 14.1·7-s − 18.3·8-s − 17.6·9-s − 71.0·10-s − 11·11-s − 36.9·12-s − 63.6·14-s − 48.4·15-s − 14.4·16-s − 5.15·17-s + 79.1·18-s + 53.7·19-s + 191.·20-s − 43.3·21-s + 49.3·22-s − 17.9·23-s + 56.1·24-s + 126.·25-s + 136.·27-s + 171.·28-s − 134.·29-s + 217.·30-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.587·3-s + 1.51·4-s + 1.41·5-s + 0.931·6-s + 0.766·7-s − 0.811·8-s − 0.654·9-s − 2.24·10-s − 0.301·11-s − 0.889·12-s − 1.21·14-s − 0.833·15-s − 0.225·16-s − 0.0735·17-s + 1.03·18-s + 0.649·19-s + 2.14·20-s − 0.450·21-s + 0.477·22-s − 0.162·23-s + 0.477·24-s + 1.00·25-s + 0.972·27-s + 1.15·28-s − 0.862·29-s + 1.32·30-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+4.48T+8T2 1 + 4.48T + 8T^{2}
3 1+3.05T+27T2 1 + 3.05T + 27T^{2}
5 1−15.8T+125T2 1 - 15.8T + 125T^{2}
7 1−14.1T+343T2 1 - 14.1T + 343T^{2}
17 1+5.15T+4.91e3T2 1 + 5.15T + 4.91e3T^{2}
19 1−53.7T+6.85e3T2 1 - 53.7T + 6.85e3T^{2}
23 1+17.9T+1.21e4T2 1 + 17.9T + 1.21e4T^{2}
29 1+134.T+2.43e4T2 1 + 134.T + 2.43e4T^{2}
31 1−43.3T+2.97e4T2 1 - 43.3T + 2.97e4T^{2}
37 1+161.T+5.06e4T2 1 + 161.T + 5.06e4T^{2}
41 1−308.T+6.89e4T2 1 - 308.T + 6.89e4T^{2}
43 1+27.9T+7.95e4T2 1 + 27.9T + 7.95e4T^{2}
47 1−139.T+1.03e5T2 1 - 139.T + 1.03e5T^{2}
53 1+165.T+1.48e5T2 1 + 165.T + 1.48e5T^{2}
59 1+542.T+2.05e5T2 1 + 542.T + 2.05e5T^{2}
61 1−655.T+2.26e5T2 1 - 655.T + 2.26e5T^{2}
67 1+969.T+3.00e5T2 1 + 969.T + 3.00e5T^{2}
71 1+174.T+3.57e5T2 1 + 174.T + 3.57e5T^{2}
73 1+1.01e3T+3.89e5T2 1 + 1.01e3T + 3.89e5T^{2}
79 1−53.3T+4.93e5T2 1 - 53.3T + 4.93e5T^{2}
83 1+1.11e3T+5.71e5T2 1 + 1.11e3T + 5.71e5T^{2}
89 1−841.T+7.04e5T2 1 - 841.T + 7.04e5T^{2}
97 1+1.20e3T+9.12e5T2 1 + 1.20e3T + 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

−8.694301456747066407399192076110, −7.85139127117387397092521209795, −7.10059876955163562285674731535, −6.09414044742117494344048091759, −5.58253031895315692166379351782, −4.66684942791016465101367318171, −2.86009734879967599207534522284, −1.93964744894277708900593668943, −1.18331237775793858311077900873, 0, 1.18331237775793858311077900873, 1.93964744894277708900593668943, 2.86009734879967599207534522284, 4.66684942791016465101367318171, 5.58253031895315692166379351782, 6.09414044742117494344048091759, 7.10059876955163562285674731535, 7.85139127117387397092521209795, 8.694301456747066407399192076110

Graph of the ZZ-function along the critical line