Properties

Label 2-1045-1.1-c5-0-225
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  − 6.79·2-s − 25.2·3-s + 14.2·4-s + 25·5-s + 171.·6-s + 60.0·7-s + 120.·8-s + 396.·9-s − 169.·10-s − 121·11-s − 359.·12-s + 944.·13-s − 407.·14-s − 632.·15-s − 1.27e3·16-s + 2.02e3·17-s − 2.69e3·18-s + 361·19-s + 355.·20-s − 1.51e3·21-s + 822.·22-s + 3.04e3·23-s − 3.05e3·24-s + 625·25-s − 6.42e3·26-s − 3.89e3·27-s + 852.·28-s + ⋯
L(s)  = 1  − 1.20·2-s − 1.62·3-s + 0.444·4-s + 0.447·5-s + 1.95·6-s + 0.462·7-s + 0.668·8-s + 1.63·9-s − 0.537·10-s − 0.301·11-s − 0.720·12-s + 1.55·13-s − 0.556·14-s − 0.725·15-s − 1.24·16-s + 1.70·17-s − 1.96·18-s + 0.229·19-s + 0.198·20-s − 0.751·21-s + 0.362·22-s + 1.19·23-s − 1.08·24-s + 0.200·25-s − 1.86·26-s − 1.02·27-s + 0.205·28-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 \( 1 + 6.79T + 32T^{2} \)
3 \( 1 + 25.2T + 243T^{2} \)
7 \( 1 - 60.0T + 1.68e4T^{2} \)
13 \( 1 - 944.T + 3.71e5T^{2} \)
17 \( 1 - 2.02e3T + 1.41e6T^{2} \)
23 \( 1 - 3.04e3T + 6.43e6T^{2} \)
29 \( 1 - 7.88e3T + 2.05e7T^{2} \)
31 \( 1 + 2.46e3T + 2.86e7T^{2} \)
37 \( 1 + 1.42e4T + 6.93e7T^{2} \)
41 \( 1 + 1.24e4T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4T + 1.47e8T^{2} \)
47 \( 1 + 1.05e4T + 2.29e8T^{2} \)
53 \( 1 - 2.42e4T + 4.18e8T^{2} \)
59 \( 1 + 3.81e4T + 7.14e8T^{2} \)
61 \( 1 + 2.85e3T + 8.44e8T^{2} \)
67 \( 1 + 5.94e4T + 1.35e9T^{2} \)
71 \( 1 - 6.54e4T + 1.80e9T^{2} \)
73 \( 1 + 1.92e4T + 2.07e9T^{2} \)
79 \( 1 - 1.10e5T + 3.07e9T^{2} \)
83 \( 1 + 8.70e4T + 3.93e9T^{2} \)
89 \( 1 + 9.15e3T + 5.58e9T^{2} \)
97 \( 1 + 4.09e4T + 8.58e9T^{2} \)
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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

−8.745303197697805597074436354674, −8.108327466370411417833341091188, −7.03997359793811321455743013027, −6.35544589438596824178754494273, −5.31680553350244359541289425524, −4.87660499522097501998065666677, −3.38281164607154314975001337471, −1.40084900723658885553828596713, −1.17320335269382657028567172254, 0, 1.17320335269382657028567172254, 1.40084900723658885553828596713, 3.38281164607154314975001337471, 4.87660499522097501998065666677, 5.31680553350244359541289425524, 6.35544589438596824178754494273, 7.03997359793811321455743013027, 8.108327466370411417833341091188, 8.745303197697805597074436354674

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