Properties

Label 2-1045-1.1-c5-0-86
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.34·2-s − 22.4·3-s + 8.30·4-s + 25·5-s + 142.·6-s + 214.·7-s + 150.·8-s + 261.·9-s − 158.·10-s + 121·11-s − 186.·12-s − 442.·13-s − 1.36e3·14-s − 561.·15-s − 1.22e3·16-s − 1.31e3·17-s − 1.65e3·18-s + 361·19-s + 207.·20-s − 4.82e3·21-s − 768.·22-s + 2.15e3·23-s − 3.37e3·24-s + 625·25-s + 2.80e3·26-s − 408.·27-s + 1.78e3·28-s + ⋯
L(s)  = 1  − 1.12·2-s − 1.44·3-s + 0.259·4-s + 0.447·5-s + 1.61·6-s + 1.65·7-s + 0.831·8-s + 1.07·9-s − 0.501·10-s + 0.301·11-s − 0.373·12-s − 0.726·13-s − 1.86·14-s − 0.644·15-s − 1.19·16-s − 1.10·17-s − 1.20·18-s + 0.229·19-s + 0.116·20-s − 2.38·21-s − 0.338·22-s + 0.849·23-s − 1.19·24-s + 0.200·25-s + 0.815·26-s − 0.107·27-s + 0.430·28-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: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8846059162\)
\(L(\frac12)\) \(\approx\) \(0.8846059162\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.34T + 32T^{2} \)
3 \( 1 + 22.4T + 243T^{2} \)
7 \( 1 - 214.T + 1.68e4T^{2} \)
13 \( 1 + 442.T + 3.71e5T^{2} \)
17 \( 1 + 1.31e3T + 1.41e6T^{2} \)
23 \( 1 - 2.15e3T + 6.43e6T^{2} \)
29 \( 1 - 5.03e3T + 2.05e7T^{2} \)
31 \( 1 + 436.T + 2.86e7T^{2} \)
37 \( 1 - 3.96e3T + 6.93e7T^{2} \)
41 \( 1 - 8.18e3T + 1.15e8T^{2} \)
43 \( 1 - 2.46e3T + 1.47e8T^{2} \)
47 \( 1 - 1.71e4T + 2.29e8T^{2} \)
53 \( 1 + 2.39e4T + 4.18e8T^{2} \)
59 \( 1 + 3.13e4T + 7.14e8T^{2} \)
61 \( 1 - 4.25e4T + 8.44e8T^{2} \)
67 \( 1 - 3.53e3T + 1.35e9T^{2} \)
71 \( 1 + 3.45e4T + 1.80e9T^{2} \)
73 \( 1 - 9.02e3T + 2.07e9T^{2} \)
79 \( 1 - 3.14e3T + 3.07e9T^{2} \)
83 \( 1 + 4.27e4T + 3.93e9T^{2} \)
89 \( 1 - 8.32e4T + 5.58e9T^{2} \)
97 \( 1 + 4.25e4T + 8.58e9T^{2} \)
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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

−9.184760582541025819694003612652, −8.479196575520496415772320039727, −7.54098521315588848970467073835, −6.83177226349728379328082783721, −5.78477195991595516016427400935, −4.72921463158147402812871368778, −4.60149579922444312708928439745, −2.29428285734113012537419210065, −1.30157158670266559592550393246, −0.60540566279666868560835605592, 0.60540566279666868560835605592, 1.30157158670266559592550393246, 2.29428285734113012537419210065, 4.60149579922444312708928439745, 4.72921463158147402812871368778, 5.78477195991595516016427400935, 6.83177226349728379328082783721, 7.54098521315588848970467073835, 8.479196575520496415772320039727, 9.184760582541025819694003612652

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