# Properties

 Degree 2 Conductor 2 Sign $1$ Motivic weight 83 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.19e12·2-s − 1.03e20·3-s + 4.83e24·4-s − 4.29e28·5-s + 2.27e32·6-s − 1.14e35·7-s − 1.06e37·8-s + 6.67e39·9-s + 9.44e40·10-s + 1.30e43·11-s − 4.99e44·12-s − 2.19e46·13-s + 2.51e47·14-s + 4.43e48·15-s + 2.33e49·16-s + 1.57e51·17-s − 1.46e52·18-s − 1.27e53·19-s − 2.07e53·20-s + 1.18e55·21-s − 2.86e55·22-s + 2.00e56·23-s + 1.09e57·24-s − 8.49e57·25-s + 4.83e58·26-s − 2.77e59·27-s − 5.52e59·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.422·5-s + 1.15·6-s − 0.968·7-s − 0.353·8-s + 1.67·9-s + 0.298·10-s + 0.787·11-s − 0.817·12-s − 1.29·13-s + 0.685·14-s + 0.690·15-s + 0.250·16-s + 1.36·17-s − 1.18·18-s − 1.08·19-s − 0.211·20-s + 1.58·21-s − 0.557·22-s + 0.616·23-s + 0.578·24-s − 0.821·25-s + 0.917·26-s − 1.10·27-s − 0.484·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+83/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2$$ $$\varepsilon$$ = $1$ motivic weight = $$83$$ character : $\chi_{2} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 2,\ (\ :83/2),\ 1)$ $L(42)$ $\approx$ $0.03413206805$ $L(\frac12)$ $\approx$ $0.03413206805$ $L(\frac{85}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + 2.19e12T$$
good3 $$1 + 1.03e20T + 3.99e39T^{2}$$
5 $$1 + 4.29e28T + 1.03e58T^{2}$$
7 $$1 + 1.14e35T + 1.39e70T^{2}$$
11 $$1 - 1.30e43T + 2.72e86T^{2}$$
13 $$1 + 2.19e46T + 2.86e92T^{2}$$
17 $$1 - 1.57e51T + 1.34e102T^{2}$$
19 $$1 + 1.27e53T + 1.36e106T^{2}$$
23 $$1 - 2.00e56T + 1.05e113T^{2}$$
29 $$1 + 5.73e60T + 2.39e121T^{2}$$
31 $$1 + 1.28e62T + 6.06e123T^{2}$$
37 $$1 + 1.04e65T + 1.44e130T^{2}$$
41 $$1 + 9.99e66T + 7.26e133T^{2}$$
43 $$1 + 9.84e67T + 3.78e135T^{2}$$
47 $$1 - 1.51e69T + 6.08e138T^{2}$$
53 $$1 - 6.81e70T + 1.30e143T^{2}$$
59 $$1 + 1.83e73T + 9.56e146T^{2}$$
61 $$1 + 2.38e74T + 1.52e148T^{2}$$
67 $$1 + 5.35e75T + 3.66e151T^{2}$$
71 $$1 - 3.92e76T + 4.51e153T^{2}$$
73 $$1 - 2.57e77T + 4.52e154T^{2}$$
79 $$1 - 1.96e78T + 3.18e157T^{2}$$
83 $$1 - 4.85e79T + 1.92e159T^{2}$$
89 $$1 + 1.41e81T + 6.30e161T^{2}$$
97 $$1 - 4.22e82T + 7.98e164T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}