# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.12·2-s − 5.79·3-s + 5.75·4-s − 18.0·6-s + 5.49·8-s + 24.5·9-s − 33.3·12-s − 2.15·13-s − 5.86·16-s + 76.7·18-s − 23·23-s − 31.8·24-s + 25·25-s − 6.72·26-s − 90.1·27-s − 13.0·29-s + 61.9·31-s − 40.3·32-s + 141.·36-s + 12.4·39-s − 66.2·41-s − 71.8·46-s + 50.9·47-s + 33.9·48-s + 49·49-s + 78.0·50-s − 12.4·52-s + ⋯
 L(s)  = 1 + 1.56·2-s − 1.93·3-s + 1.43·4-s − 3.01·6-s + 0.686·8-s + 2.72·9-s − 2.78·12-s − 0.165·13-s − 0.366·16-s + 4.26·18-s − 23-s − 1.32·24-s + 25-s − 0.258·26-s − 3.33·27-s − 0.450·29-s + 1.99·31-s − 1.25·32-s + 3.92·36-s + 0.319·39-s − 1.61·41-s − 1.56·46-s + 1.08·47-s + 0.708·48-s + 0.999·49-s + 1.56·50-s − 0.238·52-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$23$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{23} (22, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 23,\ (\ :1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $1.15945$ $L(\frac12)$ $\approx$ $1.15945$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 23$, $$F_p$$ is a polynomial of degree 2. If $p = 23$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad23 $$1 + 23T$$
good2 $$1 - 3.12T + 4T^{2}$$
3 $$1 + 5.79T + 9T^{2}$$
5 $$1 - 25T^{2}$$
7 $$1 - 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 + 2.15T + 169T^{2}$$
17 $$1 - 289T^{2}$$
19 $$1 - 361T^{2}$$
29 $$1 + 13.0T + 841T^{2}$$
31 $$1 - 61.9T + 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 + 66.2T + 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 - 50.9T + 2.20e3T^{2}$$
53 $$1 - 2.80e3T^{2}$$
59 $$1 - 26T + 3.48e3T^{2}$$
61 $$1 - 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 + 55.2T + 5.04e3T^{2}$$
73 $$1 + 88.0T + 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 - 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 - 9.40e3T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}