Properties

 Degree 2 Conductor 43 Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 3.15·2-s + 7.20·3-s + 1.96·4-s + 1.36·5-s − 22.7·6-s + 13.0·7-s + 19.0·8-s + 24.9·9-s − 4.30·10-s + 64.7·11-s + 14.1·12-s − 19.2·13-s − 41.0·14-s + 9.83·15-s − 75.8·16-s − 54.1·17-s − 78.8·18-s − 69.0·19-s + 2.67·20-s + 93.8·21-s − 204.·22-s + 29.6·23-s + 137.·24-s − 123.·25-s + 60.9·26-s − 14.6·27-s + 25.5·28-s + ⋯
 L(s)  = 1 − 1.11·2-s + 1.38·3-s + 0.245·4-s + 0.121·5-s − 1.54·6-s + 0.702·7-s + 0.842·8-s + 0.924·9-s − 0.136·10-s + 1.77·11-s + 0.340·12-s − 0.411·13-s − 0.784·14-s + 0.169·15-s − 1.18·16-s − 0.772·17-s − 1.03·18-s − 0.833·19-s + 0.0299·20-s + 0.974·21-s − 1.98·22-s + 0.268·23-s + 1.16·24-s − 0.985·25-s + 0.459·26-s − 0.104·27-s + 0.172·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{43} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$1.18786$$ $$L(\frac12)$$ $$\approx$$ $$1.18786$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 $$1 + 43T$$
good2 $$1 + 3.15T + 8T^{2}$$
3 $$1 - 7.20T + 27T^{2}$$
5 $$1 - 1.36T + 125T^{2}$$
7 $$1 - 13.0T + 343T^{2}$$
11 $$1 - 64.7T + 1.33e3T^{2}$$
13 $$1 + 19.2T + 2.19e3T^{2}$$
17 $$1 + 54.1T + 4.91e3T^{2}$$
19 $$1 + 69.0T + 6.85e3T^{2}$$
23 $$1 - 29.6T + 1.21e4T^{2}$$
29 $$1 - 13.1T + 2.43e4T^{2}$$
31 $$1 - 185.T + 2.97e4T^{2}$$
37 $$1 + 369.T + 5.06e4T^{2}$$
41 $$1 + 294.T + 6.89e4T^{2}$$
47 $$1 - 367.T + 1.03e5T^{2}$$
53 $$1 - 708.T + 1.48e5T^{2}$$
59 $$1 - 116.T + 2.05e5T^{2}$$
61 $$1 - 218.T + 2.26e5T^{2}$$
67 $$1 + 133.T + 3.00e5T^{2}$$
71 $$1 + 926.T + 3.57e5T^{2}$$
73 $$1 - 455.T + 3.89e5T^{2}$$
79 $$1 + 620.T + 4.93e5T^{2}$$
83 $$1 + 1.31e3T + 5.71e5T^{2}$$
89 $$1 - 509.T + 7.04e5T^{2}$$
97 $$1 - 965.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}