Properties

Label 2-1045-1.1-c5-0-29
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  − 10.3·2-s + 4.60·3-s + 76.1·4-s + 25·5-s − 47.8·6-s + 17.5·7-s − 459.·8-s − 221.·9-s − 259.·10-s + 121·11-s + 350.·12-s − 1.17e3·13-s − 182.·14-s + 115.·15-s + 2.33e3·16-s + 523.·17-s + 2.30e3·18-s + 361·19-s + 1.90e3·20-s + 80.7·21-s − 1.25e3·22-s − 1.60e3·23-s − 2.11e3·24-s + 625·25-s + 1.22e4·26-s − 2.14e3·27-s + 1.33e3·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.295·3-s + 2.37·4-s + 0.447·5-s − 0.543·6-s + 0.135·7-s − 2.53·8-s − 0.912·9-s − 0.822·10-s + 0.301·11-s + 0.703·12-s − 1.93·13-s − 0.248·14-s + 0.132·15-s + 2.28·16-s + 0.439·17-s + 1.67·18-s + 0.229·19-s + 1.06·20-s + 0.0399·21-s − 0.554·22-s − 0.631·23-s − 0.749·24-s + 0.200·25-s + 3.55·26-s − 0.565·27-s + 0.321·28-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

\(L(3)\) \(\approx\) \(0.3899389628\)
\(L(\frac12)\) \(\approx\) \(0.3899389628\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.3T + 32T^{2} \)
3 \( 1 - 4.60T + 243T^{2} \)
7 \( 1 - 17.5T + 1.68e4T^{2} \)
13 \( 1 + 1.17e3T + 3.71e5T^{2} \)
17 \( 1 - 523.T + 1.41e6T^{2} \)
23 \( 1 + 1.60e3T + 6.43e6T^{2} \)
29 \( 1 + 8.61e3T + 2.05e7T^{2} \)
31 \( 1 + 6.21e3T + 2.86e7T^{2} \)
37 \( 1 + 1.11e4T + 6.93e7T^{2} \)
41 \( 1 + 9.08e3T + 1.15e8T^{2} \)
43 \( 1 - 2.34e4T + 1.47e8T^{2} \)
47 \( 1 + 1.15e3T + 2.29e8T^{2} \)
53 \( 1 + 7.33e3T + 4.18e8T^{2} \)
59 \( 1 + 1.38e4T + 7.14e8T^{2} \)
61 \( 1 - 1.89e4T + 8.44e8T^{2} \)
67 \( 1 - 5.38e4T + 1.35e9T^{2} \)
71 \( 1 + 6.52e4T + 1.80e9T^{2} \)
73 \( 1 - 2.52e4T + 2.07e9T^{2} \)
79 \( 1 + 3.77e4T + 3.07e9T^{2} \)
83 \( 1 - 1.35e4T + 3.93e9T^{2} \)
89 \( 1 - 9.34e4T + 5.58e9T^{2} \)
97 \( 1 + 4.15e4T + 8.58e9T^{2} \)
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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

−9.320114169524297571310808589707, −8.526335427834919467833237964691, −7.58117141362683234383147525781, −7.22704163022056126974196380431, −6.03207918501057152946005055239, −5.20150059154732683376872213721, −3.41401400368630117623587558362, −2.33090988132671223003979637199, −1.76724402952007549674965010400, −0.33301670359579470060200488665, 0.33301670359579470060200488665, 1.76724402952007549674965010400, 2.33090988132671223003979637199, 3.41401400368630117623587558362, 5.20150059154732683376872213721, 6.03207918501057152946005055239, 7.22704163022056126974196380431, 7.58117141362683234383147525781, 8.526335427834919467833237964691, 9.320114169524297571310808589707

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