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Properties

Label 2-985-1.1-c1-0-30
Degree 22
Conductor 985985
Sign −1-1
Analytic cond. 7.865267.86526
Root an. cond. 2.804502.80450
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.236·2-s − 2.87·3-s − 1.94·4-s − 5-s − 0.677·6-s + 1.02·7-s − 0.931·8-s + 5.24·9-s − 0.236·10-s + 0.916·11-s + 5.58·12-s + 4.99·13-s + 0.241·14-s + 2.87·15-s + 3.66·16-s + 0.345·17-s + 1.23·18-s − 5.30·19-s + 1.94·20-s − 2.93·21-s + 0.216·22-s − 7.21·23-s + 2.67·24-s + 25-s + 1.17·26-s − 6.44·27-s − 1.98·28-s + ⋯
L(s)  = 1  + 0.166·2-s − 1.65·3-s − 0.972·4-s − 0.447·5-s − 0.276·6-s + 0.386·7-s − 0.329·8-s + 1.74·9-s − 0.0746·10-s + 0.276·11-s + 1.61·12-s + 1.38·13-s + 0.0644·14-s + 0.741·15-s + 0.917·16-s + 0.0838·17-s + 0.291·18-s − 1.21·19-s + 0.434·20-s − 0.640·21-s + 0.0461·22-s − 1.50·23-s + 0.545·24-s + 0.200·25-s + 0.231·26-s − 1.23·27-s − 0.375·28-s + ⋯

Functional equation

Λ(s)=(985s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(985s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 985985    =    5⋅1975 \cdot 197
Sign: −1-1
Analytic conductor: 7.865267.86526
Root analytic conductor: 2.804502.80450
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 985, ( :1/2), −1)(2,\ 985,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad5 1+T 1 + T
197 1+T 1 + T
good2 1−0.236T+2T2 1 - 0.236T + 2T^{2}
3 1+2.87T+3T2 1 + 2.87T + 3T^{2}
7 1−1.02T+7T2 1 - 1.02T + 7T^{2}
11 1−0.916T+11T2 1 - 0.916T + 11T^{2}
13 1−4.99T+13T2 1 - 4.99T + 13T^{2}
17 1−0.345T+17T2 1 - 0.345T + 17T^{2}
19 1+5.30T+19T2 1 + 5.30T + 19T^{2}
23 1+7.21T+23T2 1 + 7.21T + 23T^{2}
29 1−4.17T+29T2 1 - 4.17T + 29T^{2}
31 1−9.99T+31T2 1 - 9.99T + 31T^{2}
37 1−2.64T+37T2 1 - 2.64T + 37T^{2}
41 1+9.80T+41T2 1 + 9.80T + 41T^{2}
43 1+12.7T+43T2 1 + 12.7T + 43T^{2}
47 1+8.43T+47T2 1 + 8.43T + 47T^{2}
53 1+3.66T+53T2 1 + 3.66T + 53T^{2}
59 1−4.19T+59T2 1 - 4.19T + 59T^{2}
61 1−2.70T+61T2 1 - 2.70T + 61T^{2}
67 1−9.72T+67T2 1 - 9.72T + 67T^{2}
71 1−0.435T+71T2 1 - 0.435T + 71T^{2}
73 1+6.14T+73T2 1 + 6.14T + 73T^{2}
79 1+4.36T+79T2 1 + 4.36T + 79T^{2}
83 1+9.26T+83T2 1 + 9.26T + 83T^{2}
89 1−14.2T+89T2 1 - 14.2T + 89T^{2}
97 1+11.0T+97T2 1 + 11.0T + 97T^{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

−9.971418702623043891700848736214, −8.452433518686175009157825461961, −8.204767947991794871108089560919, −6.50687939541780888295466074856, −6.23674441840629068922137331081, −5.08210146811284364005604198164, −4.47715637092761514735997251787, −3.63547080183708698198608749544, −1.33538730130272320767838242290, 0, 1.33538730130272320767838242290, 3.63547080183708698198608749544, 4.47715637092761514735997251787, 5.08210146811284364005604198164, 6.23674441840629068922137331081, 6.50687939541780888295466074856, 8.204767947991794871108089560919, 8.452433518686175009157825461961, 9.971418702623043891700848736214

Graph of the ZZ-function along the critical line