Properties

Label 2-983-1.1-c1-0-37
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  − 2.57·2-s − 2.39·3-s + 4.62·4-s + 1.19·5-s + 6.15·6-s − 1.20·7-s − 6.74·8-s + 2.71·9-s − 3.06·10-s + 3.73·11-s − 11.0·12-s − 1.79·13-s + 3.08·14-s − 2.84·15-s + 8.11·16-s − 2.65·17-s − 6.98·18-s − 3.32·19-s + 5.50·20-s + 2.87·21-s − 9.59·22-s + 0.361·23-s + 16.1·24-s − 3.58·25-s + 4.62·26-s + 0.681·27-s − 5.54·28-s + ⋯
L(s)  = 1  − 1.81·2-s − 1.38·3-s + 2.31·4-s + 0.532·5-s + 2.51·6-s − 0.453·7-s − 2.38·8-s + 0.904·9-s − 0.968·10-s + 1.12·11-s − 3.18·12-s − 0.498·13-s + 0.825·14-s − 0.734·15-s + 2.02·16-s − 0.644·17-s − 1.64·18-s − 0.761·19-s + 1.23·20-s + 0.626·21-s − 2.04·22-s + 0.0754·23-s + 3.29·24-s − 0.716·25-s + 0.906·26-s + 0.131·27-s − 1.04·28-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 \( 1 + 2.57T + 2T^{2} \)
3 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 - 1.19T + 5T^{2} \)
7 \( 1 + 1.20T + 7T^{2} \)
11 \( 1 - 3.73T + 11T^{2} \)
13 \( 1 + 1.79T + 13T^{2} \)
17 \( 1 + 2.65T + 17T^{2} \)
19 \( 1 + 3.32T + 19T^{2} \)
23 \( 1 - 0.361T + 23T^{2} \)
29 \( 1 + 2.52T + 29T^{2} \)
31 \( 1 - 8.19T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 7.00T + 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 2.35T + 53T^{2} \)
59 \( 1 - 4.97T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 1.96T + 67T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 + 4.26T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 8.62T + 83T^{2} \)
89 \( 1 - 5.68T + 89T^{2} \)
97 \( 1 + 0.965T + 97T^{2} \)
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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

−9.642233698767781798069656852165, −9.017567809057348246605787294118, −8.018294529783912993948332528847, −6.88595705529730693894436719318, −6.44285889963405495918632039872, −5.77874021379479364667272829915, −4.34649356914268462077940282871, −2.51548194845765378513365395689, −1.27550483929408857074147591642, 0, 1.27550483929408857074147591642, 2.51548194845765378513365395689, 4.34649356914268462077940282871, 5.77874021379479364667272829915, 6.44285889963405495918632039872, 6.88595705529730693894436719318, 8.018294529783912993948332528847, 9.017567809057348246605787294118, 9.642233698767781798069656852165

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