Properties

Label 2-983-1.1-c1-0-28
Degree $2$
Conductor $983$
Sign $-1$
Analytic cond. $7.84929$
Root an. cond. $2.80165$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.05·2-s − 2.81·3-s + 2.23·4-s + 0.650·5-s + 5.80·6-s + 0.324·7-s − 0.481·8-s + 4.95·9-s − 1.33·10-s − 3.65·11-s − 6.29·12-s − 4.76·13-s − 0.667·14-s − 1.83·15-s − 3.47·16-s − 1.11·17-s − 10.1·18-s + 7.62·19-s + 1.45·20-s − 0.914·21-s + 7.51·22-s + 5.96·23-s + 1.35·24-s − 4.57·25-s + 9.80·26-s − 5.50·27-s + 0.724·28-s + ⋯
L(s)  = 1  − 1.45·2-s − 1.62·3-s + 1.11·4-s + 0.290·5-s + 2.36·6-s + 0.122·7-s − 0.170·8-s + 1.65·9-s − 0.423·10-s − 1.10·11-s − 1.81·12-s − 1.32·13-s − 0.178·14-s − 0.473·15-s − 0.869·16-s − 0.269·17-s − 2.40·18-s + 1.74·19-s + 0.324·20-s − 0.199·21-s + 1.60·22-s + 1.24·23-s + 0.276·24-s − 0.915·25-s + 1.92·26-s − 1.05·27-s + 0.136·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(983\)
Sign: $-1$
Analytic conductor: \(7.84929\)
Root analytic conductor: \(2.80165\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 983,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( 1 + T \)
good2 \( 1 + 2.05T + 2T^{2} \)
3 \( 1 + 2.81T + 3T^{2} \)
5 \( 1 - 0.650T + 5T^{2} \)
7 \( 1 - 0.324T + 7T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + 4.76T + 13T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 - 7.62T + 19T^{2} \)
23 \( 1 - 5.96T + 23T^{2} \)
29 \( 1 - 9.04T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 + 3.84T + 37T^{2} \)
41 \( 1 - 4.18T + 41T^{2} \)
43 \( 1 - 6.53T + 43T^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 5.03T + 61T^{2} \)
67 \( 1 + 9.87T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 4.36T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.918838396838255666214176267128, −8.950771470628264846059979069826, −7.67939209306844669573499758927, −7.30532952048052752031056251093, −6.31676641856174550808039122246, −5.20704343852167339938916886152, −4.78976946928943323053628360494, −2.65740311129028304119319437105, −1.18514791276955213778956902876, 0, 1.18514791276955213778956902876, 2.65740311129028304119319437105, 4.78976946928943323053628360494, 5.20704343852167339938916886152, 6.31676641856174550808039122246, 7.30532952048052752031056251093, 7.67939209306844669573499758927, 8.950771470628264846059979069826, 9.918838396838255666214176267128

Graph of the $Z$-function along the critical line