Properties

Label 2-1045-1.1-c5-0-248
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.95·2-s + 14.0·3-s + 31.2·4-s + 25·5-s − 111.·6-s + 100.·7-s + 6.27·8-s − 45.7·9-s − 198.·10-s + 121·11-s + 438.·12-s + 570.·13-s − 801.·14-s + 351.·15-s − 1.04e3·16-s − 1.18e3·17-s + 363.·18-s − 361·19-s + 780.·20-s + 1.41e3·21-s − 962.·22-s + 3.60e3·23-s + 88.1·24-s + 625·25-s − 4.53e3·26-s − 4.05e3·27-s + 3.14e3·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.900·3-s + 0.975·4-s + 0.447·5-s − 1.26·6-s + 0.777·7-s + 0.0346·8-s − 0.188·9-s − 0.628·10-s + 0.301·11-s + 0.878·12-s + 0.936·13-s − 1.09·14-s + 0.402·15-s − 1.02·16-s − 0.990·17-s + 0.264·18-s − 0.229·19-s + 0.436·20-s + 0.700·21-s − 0.423·22-s + 1.42·23-s + 0.0312·24-s + 0.200·25-s − 1.31·26-s − 1.07·27-s + 0.758·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25T \)
11 \( 1 - 121T \)
19 \( 1 + 361T \)
good2 \( 1 + 7.95T + 32T^{2} \)
3 \( 1 - 14.0T + 243T^{2} \)
7 \( 1 - 100.T + 1.68e4T^{2} \)
13 \( 1 - 570.T + 3.71e5T^{2} \)
17 \( 1 + 1.18e3T + 1.41e6T^{2} \)
23 \( 1 - 3.60e3T + 6.43e6T^{2} \)
29 \( 1 + 1.10e3T + 2.05e7T^{2} \)
31 \( 1 + 3.96e3T + 2.86e7T^{2} \)
37 \( 1 + 9.36e3T + 6.93e7T^{2} \)
41 \( 1 + 1.14e4T + 1.15e8T^{2} \)
43 \( 1 - 1.60e3T + 1.47e8T^{2} \)
47 \( 1 - 617.T + 2.29e8T^{2} \)
53 \( 1 - 2.08e4T + 4.18e8T^{2} \)
59 \( 1 + 3.42e4T + 7.14e8T^{2} \)
61 \( 1 + 8.29e3T + 8.44e8T^{2} \)
67 \( 1 + 1.14e4T + 1.35e9T^{2} \)
71 \( 1 + 2.69e4T + 1.80e9T^{2} \)
73 \( 1 - 6.89e4T + 2.07e9T^{2} \)
79 \( 1 + 8.12e4T + 3.07e9T^{2} \)
83 \( 1 + 9.69e4T + 3.93e9T^{2} \)
89 \( 1 + 1.48e5T + 5.58e9T^{2} \)
97 \( 1 + 1.70e5T + 8.58e9T^{2} \)
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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

−8.751918337656831252595771186909, −8.396781280320595336439422724807, −7.39289248212774971659398149406, −6.64767768385755558402614835680, −5.41400178641222098222336631341, −4.27277328248998444186851448953, −3.05086100793223957167795583032, −1.94102796326827207030161141828, −1.35110140372277150722634477740, 0, 1.35110140372277150722634477740, 1.94102796326827207030161141828, 3.05086100793223957167795583032, 4.27277328248998444186851448953, 5.41400178641222098222336631341, 6.64767768385755558402614835680, 7.39289248212774971659398149406, 8.396781280320595336439422724807, 8.751918337656831252595771186909

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