# Properties

 Degree 2 Conductor 23 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.61·2-s + 2.23·3-s + 0.618·4-s − 3.23·5-s − 3.61·6-s − 1.23·7-s + 2.23·8-s + 2.00·9-s + 5.23·10-s − 0.763·11-s + 1.38·12-s + 3·13-s + 2.00·14-s − 7.23·15-s − 4.85·16-s + 5.23·17-s − 3.23·18-s − 2·19-s − 2.00·20-s − 2.76·21-s + 1.23·22-s + 23-s + 5.00·24-s + 5.47·25-s − 4.85·26-s − 2.23·27-s − 0.763·28-s + ⋯
 L(s)  = 1 − 1.14·2-s + 1.29·3-s + 0.309·4-s − 1.44·5-s − 1.47·6-s − 0.467·7-s + 0.790·8-s + 0.666·9-s + 1.65·10-s − 0.230·11-s + 0.398·12-s + 0.832·13-s + 0.534·14-s − 1.86·15-s − 1.21·16-s + 1.26·17-s − 0.762·18-s − 0.458·19-s − 0.447·20-s − 0.603·21-s + 0.263·22-s + 0.208·23-s + 1.02·24-s + 1.09·25-s − 0.951·26-s − 0.430·27-s − 0.144·28-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$23$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{23} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 23,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.450379$ $L(\frac12)$ $\approx$ $0.450379$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 23$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 23$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad23 $$1 - T$$
good2 $$1 + 1.61T + 2T^{2}$$
3 $$1 - 2.23T + 3T^{2}$$
5 $$1 + 3.23T + 5T^{2}$$
7 $$1 + 1.23T + 7T^{2}$$
11 $$1 + 0.763T + 11T^{2}$$
13 $$1 - 3T + 13T^{2}$$
17 $$1 - 5.23T + 17T^{2}$$
19 $$1 + 2T + 19T^{2}$$
29 $$1 + 3T + 29T^{2}$$
31 $$1 + 6.70T + 31T^{2}$$
37 $$1 - 3.23T + 37T^{2}$$
41 $$1 - 5.47T + 41T^{2}$$
43 $$1 + 43T^{2}$$
47 $$1 - 2.23T + 47T^{2}$$
53 $$1 + 8.47T + 53T^{2}$$
59 $$1 + 2.47T + 59T^{2}$$
61 $$1 - 10.9T + 61T^{2}$$
67 $$1 + 7.23T + 67T^{2}$$
71 $$1 - 7.76T + 71T^{2}$$
73 $$1 - 15.4T + 73T^{2}$$
79 $$1 - 6.94T + 79T^{2}$$
83 $$1 + 13.2T + 83T^{2}$$
89 $$1 + 1.52T + 89T^{2}$$
97 $$1 - 4.29T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}