Properties

Label 2-1045-1.1-c5-0-144
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  − 8.88·2-s − 24.9·3-s + 46.9·4-s + 25·5-s + 221.·6-s − 188.·7-s − 132.·8-s + 378.·9-s − 222.·10-s + 121·11-s − 1.17e3·12-s + 1.01e3·13-s + 1.67e3·14-s − 623.·15-s − 322.·16-s + 136.·17-s − 3.36e3·18-s − 361·19-s + 1.17e3·20-s + 4.69e3·21-s − 1.07e3·22-s − 2.49e3·23-s + 3.31e3·24-s + 625·25-s − 9.00e3·26-s − 3.38e3·27-s − 8.83e3·28-s + ⋯
L(s)  = 1  − 1.57·2-s − 1.59·3-s + 1.46·4-s + 0.447·5-s + 2.51·6-s − 1.45·7-s − 0.733·8-s + 1.55·9-s − 0.702·10-s + 0.301·11-s − 2.34·12-s + 1.66·13-s + 2.27·14-s − 0.715·15-s − 0.315·16-s + 0.114·17-s − 2.44·18-s − 0.229·19-s + 0.656·20-s + 2.32·21-s − 0.473·22-s − 0.985·23-s + 1.17·24-s + 0.200·25-s − 2.61·26-s − 0.893·27-s − 2.12·28-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 \( 1 + 8.88T + 32T^{2} \)
3 \( 1 + 24.9T + 243T^{2} \)
7 \( 1 + 188.T + 1.68e4T^{2} \)
13 \( 1 - 1.01e3T + 3.71e5T^{2} \)
17 \( 1 - 136.T + 1.41e6T^{2} \)
23 \( 1 + 2.49e3T + 6.43e6T^{2} \)
29 \( 1 - 6.92e3T + 2.05e7T^{2} \)
31 \( 1 - 1.84e3T + 2.86e7T^{2} \)
37 \( 1 + 1.00e4T + 6.93e7T^{2} \)
41 \( 1 - 1.18e4T + 1.15e8T^{2} \)
43 \( 1 + 3.91e3T + 1.47e8T^{2} \)
47 \( 1 + 1.64e4T + 2.29e8T^{2} \)
53 \( 1 + 3.59e4T + 4.18e8T^{2} \)
59 \( 1 + 2.06e4T + 7.14e8T^{2} \)
61 \( 1 + 2.22e3T + 8.44e8T^{2} \)
67 \( 1 + 2.92e4T + 1.35e9T^{2} \)
71 \( 1 + 2.55e4T + 1.80e9T^{2} \)
73 \( 1 - 3.14e4T + 2.07e9T^{2} \)
79 \( 1 + 9.28e4T + 3.07e9T^{2} \)
83 \( 1 - 8.88e4T + 3.93e9T^{2} \)
89 \( 1 - 1.27e5T + 5.58e9T^{2} \)
97 \( 1 + 3.52e4T + 8.58e9T^{2} \)
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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

−8.978815555006974457327578397093, −8.094025848832087534799705139502, −6.84262170369698080602520039080, −6.32710899841742054928709426930, −5.96288937370534713089913766972, −4.53832444692176103161822825110, −3.22552245021680983921595714724, −1.65262269896172255324525441049, −0.814245287458455770463435803831, 0, 0.814245287458455770463435803831, 1.65262269896172255324525441049, 3.22552245021680983921595714724, 4.53832444692176103161822825110, 5.96288937370534713089913766972, 6.32710899841742054928709426930, 6.84262170369698080602520039080, 8.094025848832087534799705139502, 8.978815555006974457327578397093

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