Properties

Label 2-1045-1.1-c5-0-93
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  − 10.1·2-s − 26.5·3-s + 71.1·4-s + 25·5-s + 269.·6-s + 225.·7-s − 397.·8-s + 461.·9-s − 253.·10-s + 121·11-s − 1.88e3·12-s + 301.·13-s − 2.28e3·14-s − 663.·15-s + 1.75e3·16-s + 2.04e3·17-s − 4.68e3·18-s + 361·19-s + 1.77e3·20-s − 5.98e3·21-s − 1.22e3·22-s − 4.32e3·23-s + 1.05e4·24-s + 625·25-s − 3.06e3·26-s − 5.80e3·27-s + 1.60e4·28-s + ⋯
L(s)  = 1  − 1.79·2-s − 1.70·3-s + 2.22·4-s + 0.447·5-s + 3.05·6-s + 1.73·7-s − 2.19·8-s + 1.89·9-s − 0.802·10-s + 0.301·11-s − 3.78·12-s + 0.494·13-s − 3.12·14-s − 0.761·15-s + 1.71·16-s + 1.71·17-s − 3.40·18-s + 0.229·19-s + 0.993·20-s − 2.96·21-s − 0.541·22-s − 1.70·23-s + 3.73·24-s + 0.200·25-s − 0.888·26-s − 1.53·27-s + 3.86·28-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

\(L(3)\) \(\approx\) \(0.7790420773\)
\(L(\frac12)\) \(\approx\) \(0.7790420773\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.1T + 32T^{2} \)
3 \( 1 + 26.5T + 243T^{2} \)
7 \( 1 - 225.T + 1.68e4T^{2} \)
13 \( 1 - 301.T + 3.71e5T^{2} \)
17 \( 1 - 2.04e3T + 1.41e6T^{2} \)
23 \( 1 + 4.32e3T + 6.43e6T^{2} \)
29 \( 1 + 3.91e3T + 2.05e7T^{2} \)
31 \( 1 + 6.97e3T + 2.86e7T^{2} \)
37 \( 1 + 5.29e3T + 6.93e7T^{2} \)
41 \( 1 - 9.35e3T + 1.15e8T^{2} \)
43 \( 1 + 2.26e3T + 1.47e8T^{2} \)
47 \( 1 - 2.01e4T + 2.29e8T^{2} \)
53 \( 1 - 9.53e3T + 4.18e8T^{2} \)
59 \( 1 - 4.91e4T + 7.14e8T^{2} \)
61 \( 1 + 5.38e4T + 8.44e8T^{2} \)
67 \( 1 - 6.75e4T + 1.35e9T^{2} \)
71 \( 1 - 3.76e4T + 1.80e9T^{2} \)
73 \( 1 - 1.02e4T + 2.07e9T^{2} \)
79 \( 1 + 8.67e4T + 3.07e9T^{2} \)
83 \( 1 - 5.27e4T + 3.93e9T^{2} \)
89 \( 1 + 3.34e4T + 5.58e9T^{2} \)
97 \( 1 + 1.25e5T + 8.58e9T^{2} \)
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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

−9.360852734749951754674576342090, −8.270735889251003710183711295786, −7.64830131482937931211641644191, −6.90331127918829556587161774412, −5.66674000450584339743156243877, −5.53645600456720606590312182864, −4.04240308226823724897266155939, −1.96530465623887057211574078263, −1.36452986468726712496967186065, −0.62719315628655841889088186893, 0.62719315628655841889088186893, 1.36452986468726712496967186065, 1.96530465623887057211574078263, 4.04240308226823724897266155939, 5.53645600456720606590312182864, 5.66674000450584339743156243877, 6.90331127918829556587161774412, 7.64830131482937931211641644191, 8.270735889251003710183711295786, 9.360852734749951754674576342090

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