Properties

Label 2-983-1.1-c1-0-43
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.05·2-s + 2.27·3-s − 0.889·4-s − 0.660·5-s + 2.40·6-s + 3.15·7-s − 3.04·8-s + 2.19·9-s − 0.696·10-s + 2.39·11-s − 2.02·12-s + 4.12·13-s + 3.32·14-s − 1.50·15-s − 1.42·16-s + 0.223·17-s + 2.30·18-s − 2.55·19-s + 0.587·20-s + 7.19·21-s + 2.51·22-s + 8.39·23-s − 6.93·24-s − 4.56·25-s + 4.35·26-s − 1.84·27-s − 2.80·28-s + ⋯
L(s)  = 1  + 0.745·2-s + 1.31·3-s − 0.444·4-s − 0.295·5-s + 0.980·6-s + 1.19·7-s − 1.07·8-s + 0.730·9-s − 0.220·10-s + 0.720·11-s − 0.585·12-s + 1.14·13-s + 0.888·14-s − 0.388·15-s − 0.357·16-s + 0.0543·17-s + 0.544·18-s − 0.586·19-s + 0.131·20-s + 1.56·21-s + 0.537·22-s + 1.75·23-s − 1.41·24-s − 0.912·25-s + 0.853·26-s − 0.354·27-s − 0.530·28-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: \(0\)
Selberg data: \((2,\ 983,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.342463669\)
\(L(\frac12)\) \(\approx\) \(3.342463669\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.05T + 2T^{2} \)
3 \( 1 - 2.27T + 3T^{2} \)
5 \( 1 + 0.660T + 5T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 - 2.39T + 11T^{2} \)
13 \( 1 - 4.12T + 13T^{2} \)
17 \( 1 - 0.223T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 - 8.39T + 23T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 + 8.46T + 31T^{2} \)
37 \( 1 - 9.00T + 37T^{2} \)
41 \( 1 - 1.14T + 41T^{2} \)
43 \( 1 + 0.428T + 43T^{2} \)
47 \( 1 + 9.60T + 47T^{2} \)
53 \( 1 + 6.65T + 53T^{2} \)
59 \( 1 - 9.50T + 59T^{2} \)
61 \( 1 + 6.35T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 9.94T + 71T^{2} \)
73 \( 1 + 3.59T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 9.35T + 97T^{2} \)
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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

−9.683826904172161190884027781544, −8.919957946285284100583425814209, −8.448880571204551881989550536802, −7.77281588044116134307468711005, −6.55449581799082001689688453440, −5.44832856484579355246405552990, −4.41730188960701406847212448942, −3.79305841733673664137787366893, −2.88034291934486793191037056494, −1.46977678712610590285330131420, 1.46977678712610590285330131420, 2.88034291934486793191037056494, 3.79305841733673664137787366893, 4.41730188960701406847212448942, 5.44832856484579355246405552990, 6.55449581799082001689688453440, 7.77281588044116134307468711005, 8.448880571204551881989550536802, 8.919957946285284100583425814209, 9.683826904172161190884027781544

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