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Properties

Label 2-1617-1.1-c3-0-187
Degree 22
Conductor 16171617
Sign −1-1
Analytic cond. 95.406095.4060
Root an. cond. 9.767609.76760
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  + 1.59·2-s + 3·3-s − 5.44·4-s + 17.2·5-s + 4.79·6-s − 21.5·8-s + 9·9-s + 27.6·10-s + 11·11-s − 16.3·12-s − 46.1·13-s + 51.8·15-s + 9.11·16-s − 19.8·17-s + 14.3·18-s − 76.5·19-s − 93.9·20-s + 17.5·22-s − 163.·23-s − 64.5·24-s + 173.·25-s − 73.7·26-s + 27·27-s + 158.·29-s + 82.9·30-s − 170.·31-s + 186.·32-s + ⋯
L(s)  = 1  + 0.565·2-s + 0.577·3-s − 0.680·4-s + 1.54·5-s + 0.326·6-s − 0.950·8-s + 0.333·9-s + 0.874·10-s + 0.301·11-s − 0.392·12-s − 0.983·13-s + 0.892·15-s + 0.142·16-s − 0.283·17-s + 0.188·18-s − 0.924·19-s − 1.05·20-s + 0.170·22-s − 1.47·23-s − 0.548·24-s + 1.38·25-s − 0.556·26-s + 0.192·27-s + 1.01·29-s + 0.504·30-s − 0.989·31-s + 1.03·32-s + ⋯

Functional equation

Λ(s)=(1617s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1617s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16171617    =    3⋅72⋅113 \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 95.406095.4060
Root analytic conductor: 9.767609.76760
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1617, ( :3/2), −1)(2,\ 1617,\ (\ :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)
bad3 1−3T 1 - 3T
7 1 1
11 1−11T 1 - 11T
good2 1−1.59T+8T2 1 - 1.59T + 8T^{2}
5 1−17.2T+125T2 1 - 17.2T + 125T^{2}
13 1+46.1T+2.19e3T2 1 + 46.1T + 2.19e3T^{2}
17 1+19.8T+4.91e3T2 1 + 19.8T + 4.91e3T^{2}
19 1+76.5T+6.85e3T2 1 + 76.5T + 6.85e3T^{2}
23 1+163.T+1.21e4T2 1 + 163.T + 1.21e4T^{2}
29 1−158.T+2.43e4T2 1 - 158.T + 2.43e4T^{2}
31 1+170.T+2.97e4T2 1 + 170.T + 2.97e4T^{2}
37 1+245.T+5.06e4T2 1 + 245.T + 5.06e4T^{2}
41 1−3.33T+6.89e4T2 1 - 3.33T + 6.89e4T^{2}
43 1+122.T+7.95e4T2 1 + 122.T + 7.95e4T^{2}
47 1+390.T+1.03e5T2 1 + 390.T + 1.03e5T^{2}
53 1+410.T+1.48e5T2 1 + 410.T + 1.48e5T^{2}
59 1+408.T+2.05e5T2 1 + 408.T + 2.05e5T^{2}
61 1−21.9T+2.26e5T2 1 - 21.9T + 2.26e5T^{2}
67 1−618.T+3.00e5T2 1 - 618.T + 3.00e5T^{2}
71 1−929.T+3.57e5T2 1 - 929.T + 3.57e5T^{2}
73 1+868.T+3.89e5T2 1 + 868.T + 3.89e5T^{2}
79 1−152.T+4.93e5T2 1 - 152.T + 4.93e5T^{2}
83 1−100.T+5.71e5T2 1 - 100.T + 5.71e5T^{2}
89 1−1.06e3T+7.04e5T2 1 - 1.06e3T + 7.04e5T^{2}
97 1+1.41e3T+9.12e5T2 1 + 1.41e3T + 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

−8.758165875451702435532292292127, −8.043187340358688915225297526636, −6.74317372745553499999873563689, −6.13289043466406624192129568756, −5.22767461222078063757164759509, −4.54686553358942437178016167905, −3.52360879028872061928551112991, −2.45059689911978263152445739519, −1.69029688114168653335619848658, 0, 1.69029688114168653335619848658, 2.45059689911978263152445739519, 3.52360879028872061928551112991, 4.54686553358942437178016167905, 5.22767461222078063757164759509, 6.13289043466406624192129568756, 6.74317372745553499999873563689, 8.043187340358688915225297526636, 8.758165875451702435532292292127

Graph of the ZZ-function along the critical line