# Properties

 Degree 2 Conductor 3 Sign $1$ Motivic weight 37 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.21e5·2-s + 3.87e8·3-s − 3.37e10·4-s + 1.53e13·5-s + 1.24e14·6-s + 2.48e15·7-s − 5.51e16·8-s + 1.50e17·9-s + 4.93e18·10-s − 1.32e18·11-s − 1.30e19·12-s + 1.53e20·13-s + 8.00e20·14-s + 5.94e21·15-s − 1.31e22·16-s + 1.04e23·17-s + 4.83e22·18-s − 6.96e23·19-s − 5.17e23·20-s + 9.63e23·21-s − 4.25e23·22-s + 1.11e25·23-s − 2.13e25·24-s + 1.62e26·25-s + 4.93e25·26-s + 5.81e25·27-s − 8.39e25·28-s + ⋯
 L(s)  = 1 + 0.868·2-s + 0.577·3-s − 0.245·4-s + 1.79·5-s + 0.501·6-s + 0.577·7-s − 1.08·8-s + 0.333·9-s + 1.56·10-s − 0.0716·11-s − 0.141·12-s + 0.378·13-s + 0.501·14-s + 1.03·15-s − 0.694·16-s + 1.80·17-s + 0.289·18-s − 1.53·19-s − 0.441·20-s + 0.333·21-s − 0.0622·22-s + 0.719·23-s − 0.624·24-s + 2.23·25-s + 0.328·26-s + 0.192·27-s − 0.141·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $1$ motivic weight = $$37$$ character : $\chi_{3} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :37/2),\ 1)$ $L(19)$ $\approx$ $4.681073251$ $L(\frac12)$ $\approx$ $4.681073251$ $L(\frac{39}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 - 3.87e8T$$
good2 $$1 - 3.21e5T + 1.37e11T^{2}$$
5 $$1 - 1.53e13T + 7.27e25T^{2}$$
7 $$1 - 2.48e15T + 1.85e31T^{2}$$
11 $$1 + 1.32e18T + 3.40e38T^{2}$$
13 $$1 - 1.53e20T + 1.64e41T^{2}$$
17 $$1 - 1.04e23T + 3.36e45T^{2}$$
19 $$1 + 6.96e23T + 2.06e47T^{2}$$
23 $$1 - 1.11e25T + 2.42e50T^{2}$$
29 $$1 - 1.43e27T + 1.28e54T^{2}$$
31 $$1 - 3.90e26T + 1.51e55T^{2}$$
37 $$1 + 8.53e28T + 1.05e58T^{2}$$
41 $$1 - 4.51e29T + 4.70e59T^{2}$$
43 $$1 + 2.04e30T + 2.74e60T^{2}$$
47 $$1 + 3.80e30T + 7.37e61T^{2}$$
53 $$1 - 2.08e31T + 6.28e63T^{2}$$
59 $$1 + 1.01e33T + 3.32e65T^{2}$$
61 $$1 + 1.04e33T + 1.14e66T^{2}$$
67 $$1 - 2.33e33T + 3.67e67T^{2}$$
71 $$1 - 1.08e33T + 3.13e68T^{2}$$
73 $$1 + 3.23e34T + 8.76e68T^{2}$$
79 $$1 + 2.35e35T + 1.63e70T^{2}$$
83 $$1 + 1.58e35T + 1.01e71T^{2}$$
89 $$1 + 3.16e35T + 1.34e72T^{2}$$
97 $$1 - 6.89e36T + 3.24e73T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}