# Properties

 Degree 2 Conductor $1$ Sign $-1$ Motivic weight 73 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.59e11·2-s + 2.55e16·3-s + 1.59e22·4-s + 2.70e25·5-s − 4.06e27·6-s − 5.83e30·7-s − 1.04e33·8-s − 6.69e34·9-s − 4.31e36·10-s + 1.33e38·11-s + 4.07e38·12-s − 6.53e40·13-s + 9.30e41·14-s + 6.91e41·15-s + 1.51e43·16-s + 1.34e45·17-s + 1.06e46·18-s + 4.71e46·19-s + 4.32e47·20-s − 1.48e47·21-s − 2.12e49·22-s − 7.28e48·23-s − 2.65e49·24-s − 3.25e50·25-s + 1.04e52·26-s − 3.43e51·27-s − 9.32e52·28-s + ⋯
 L(s)  = 1 − 1.64·2-s + 0.0981·3-s + 1.69·4-s + 0.832·5-s − 0.161·6-s − 0.832·7-s − 1.13·8-s − 0.990·9-s − 1.36·10-s + 1.29·11-s + 0.166·12-s − 1.43·13-s + 1.36·14-s + 0.0817·15-s + 0.169·16-s + 1.65·17-s + 1.62·18-s + 0.997·19-s + 1.40·20-s − 0.0816·21-s − 2.13·22-s − 0.144·23-s − 0.111·24-s − 0.307·25-s + 2.35·26-s − 0.195·27-s − 1.40·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1$$ $$\varepsilon$$ = $-1$ motivic weight = $$73$$ character : $\chi_{1} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 1,\ (\ :73/2),\ -1)$$ $$L(37)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{75}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where,$$F_p(T)$$ is a polynomial of degree 2.
$p$$F_p(T)$
good2 $$1 + 1.59e11T + 9.44e21T^{2}$$
3 $$1 - 2.55e16T + 6.75e34T^{2}$$
5 $$1 - 2.70e25T + 1.05e51T^{2}$$
7 $$1 + 5.83e30T + 4.92e61T^{2}$$
11 $$1 - 1.33e38T + 1.05e76T^{2}$$
13 $$1 + 6.53e40T + 2.07e81T^{2}$$
17 $$1 - 1.34e45T + 6.64e89T^{2}$$
19 $$1 - 4.71e46T + 2.23e93T^{2}$$
23 $$1 + 7.28e48T + 2.54e99T^{2}$$
29 $$1 + 1.30e53T + 5.68e106T^{2}$$
31 $$1 + 6.62e53T + 7.40e108T^{2}$$
37 $$1 - 1.59e57T + 3.01e114T^{2}$$
41 $$1 + 1.09e59T + 5.41e117T^{2}$$
43 $$1 - 6.52e59T + 1.75e119T^{2}$$
47 $$1 - 6.06e60T + 1.15e122T^{2}$$
53 $$1 + 1.02e63T + 7.44e125T^{2}$$
59 $$1 + 2.47e64T + 1.87e129T^{2}$$
61 $$1 + 8.75e64T + 2.13e130T^{2}$$
67 $$1 + 2.58e66T + 2.01e133T^{2}$$
71 $$1 + 4.91e67T + 1.38e135T^{2}$$
73 $$1 + 1.02e68T + 1.05e136T^{2}$$
79 $$1 + 2.22e69T + 3.36e138T^{2}$$
83 $$1 + 1.44e70T + 1.23e140T^{2}$$
89 $$1 - 3.70e70T + 2.02e142T^{2}$$
97 $$1 - 3.38e71T + 1.08e145T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}