# Properties

 Label 2-23-1.1-c5-0-8 Degree $2$ Conductor $23$ Sign $-1$ Analytic cond. $3.68882$ Root an. cond. $1.92063$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.31·2-s − 27.6·3-s − 3.75·4-s − 23.8·5-s − 147.·6-s − 11.7·7-s − 190.·8-s + 522.·9-s − 126.·10-s + 280.·11-s + 103.·12-s − 835.·13-s − 62.6·14-s + 660.·15-s − 889.·16-s − 1.80e3·17-s + 2.77e3·18-s + 2.35e3·19-s + 89.6·20-s + 325.·21-s + 1.49e3·22-s − 529·23-s + 5.25e3·24-s − 2.55e3·25-s − 4.43e3·26-s − 7.72e3·27-s + 44.2·28-s + ⋯
 L(s)  = 1 + 0.939·2-s − 1.77·3-s − 0.117·4-s − 0.427·5-s − 1.66·6-s − 0.0908·7-s − 1.04·8-s + 2.14·9-s − 0.401·10-s + 0.699·11-s + 0.208·12-s − 1.37·13-s − 0.0854·14-s + 0.757·15-s − 0.868·16-s − 1.51·17-s + 2.01·18-s + 1.49·19-s + 0.0501·20-s + 0.161·21-s + 0.657·22-s − 0.208·23-s + 1.86·24-s − 0.817·25-s − 1.28·26-s − 2.03·27-s + 0.0106·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$23$$ Sign: $-1$ Analytic conductor: $$3.68882$$ Root analytic conductor: $$1.92063$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 23,\ (\ :5/2),\ -1)$$

## 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$$F_p(T)$
bad23 $$1 + 529T$$
good2 $$1 - 5.31T + 32T^{2}$$
3 $$1 + 27.6T + 243T^{2}$$
5 $$1 + 23.8T + 3.12e3T^{2}$$
7 $$1 + 11.7T + 1.68e4T^{2}$$
11 $$1 - 280.T + 1.61e5T^{2}$$
13 $$1 + 835.T + 3.71e5T^{2}$$
17 $$1 + 1.80e3T + 1.41e6T^{2}$$
19 $$1 - 2.35e3T + 2.47e6T^{2}$$
29 $$1 + 1.20e3T + 2.05e7T^{2}$$
31 $$1 + 3.39e3T + 2.86e7T^{2}$$
37 $$1 - 8.35e3T + 6.93e7T^{2}$$
41 $$1 + 1.07e4T + 1.15e8T^{2}$$
43 $$1 + 803.T + 1.47e8T^{2}$$
47 $$1 + 4.89e3T + 2.29e8T^{2}$$
53 $$1 - 3.93e4T + 4.18e8T^{2}$$
59 $$1 - 3.95e4T + 7.14e8T^{2}$$
61 $$1 + 1.53e4T + 8.44e8T^{2}$$
67 $$1 + 6.07e4T + 1.35e9T^{2}$$
71 $$1 - 3.91e4T + 1.80e9T^{2}$$
73 $$1 + 5.87e3T + 2.07e9T^{2}$$
79 $$1 + 6.36e4T + 3.07e9T^{2}$$
83 $$1 + 6.36e4T + 3.93e9T^{2}$$
89 $$1 - 3.62e4T + 5.58e9T^{2}$$
97 $$1 + 6.41e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$