Properties

Label 2-1045-1.1-c5-0-118
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  − 8.31·2-s + 15.2·3-s + 37.0·4-s + 25·5-s − 126.·6-s + 144.·7-s − 42.3·8-s − 9.68·9-s − 207.·10-s + 121·11-s + 566.·12-s + 149.·13-s − 1.20e3·14-s + 381.·15-s − 834.·16-s − 452.·17-s + 80.5·18-s + 361·19-s + 927.·20-s + 2.20e3·21-s − 1.00e3·22-s − 1.57e3·23-s − 647.·24-s + 625·25-s − 1.24e3·26-s − 3.85e3·27-s + 5.36e3·28-s + ⋯
L(s)  = 1  − 1.46·2-s + 0.979·3-s + 1.15·4-s + 0.447·5-s − 1.43·6-s + 1.11·7-s − 0.234·8-s − 0.0398·9-s − 0.657·10-s + 0.301·11-s + 1.13·12-s + 0.245·13-s − 1.63·14-s + 0.438·15-s − 0.815·16-s − 0.379·17-s + 0.0585·18-s + 0.229·19-s + 0.518·20-s + 1.09·21-s − 0.443·22-s − 0.621·23-s − 0.229·24-s + 0.200·25-s − 0.360·26-s − 1.01·27-s + 1.29·28-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

\(L(3)\) \(\approx\) \(1.876009533\)
\(L(\frac12)\) \(\approx\) \(1.876009533\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.31T + 32T^{2} \)
3 \( 1 - 15.2T + 243T^{2} \)
7 \( 1 - 144.T + 1.68e4T^{2} \)
13 \( 1 - 149.T + 3.71e5T^{2} \)
17 \( 1 + 452.T + 1.41e6T^{2} \)
23 \( 1 + 1.57e3T + 6.43e6T^{2} \)
29 \( 1 + 5.58e3T + 2.05e7T^{2} \)
31 \( 1 - 3.52e3T + 2.86e7T^{2} \)
37 \( 1 - 5.39e3T + 6.93e7T^{2} \)
41 \( 1 + 4.22e3T + 1.15e8T^{2} \)
43 \( 1 + 1.58e4T + 1.47e8T^{2} \)
47 \( 1 - 2.21e4T + 2.29e8T^{2} \)
53 \( 1 - 1.44e4T + 4.18e8T^{2} \)
59 \( 1 - 4.97e4T + 7.14e8T^{2} \)
61 \( 1 + 1.77e4T + 8.44e8T^{2} \)
67 \( 1 + 6.24e4T + 1.35e9T^{2} \)
71 \( 1 - 2.41e4T + 1.80e9T^{2} \)
73 \( 1 - 5.46e4T + 2.07e9T^{2} \)
79 \( 1 - 9.63e4T + 3.07e9T^{2} \)
83 \( 1 - 6.64e4T + 3.93e9T^{2} \)
89 \( 1 - 1.05e5T + 5.58e9T^{2} \)
97 \( 1 - 2.55e4T + 8.58e9T^{2} \)
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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

−9.080779187203271188878279256727, −8.438438900544740038654593074738, −7.923134922421339507358138233549, −7.15345684869585515721249639217, −6.02004582245004503188550851241, −4.84038502493803003057817095929, −3.67931472655228187221136752970, −2.30162136337090220682563590610, −1.80244507703818963572309383154, −0.71527450103181445538246229906, 0.71527450103181445538246229906, 1.80244507703818963572309383154, 2.30162136337090220682563590610, 3.67931472655228187221136752970, 4.84038502493803003057817095929, 6.02004582245004503188550851241, 7.15345684869585515721249639217, 7.923134922421339507358138233549, 8.438438900544740038654593074738, 9.080779187203271188878279256727

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