# Properties

 Degree 2 Conductor 41 Sign $0.994 - 0.104i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.698 + 0.227i)2-s + (0.0432 + 0.0432i)3-s + (−1.18 − 0.858i)4-s + (−0.422 + 0.581i)5-s + (0.0204 + 0.0400i)6-s + (−1.42 + 2.79i)7-s + (−1.49 − 2.05i)8-s − 2.99i·9-s + (−0.426 + 0.310i)10-s + (3.06 + 0.486i)11-s + (−0.0139 − 0.0883i)12-s + (−1.79 + 0.912i)13-s + (−1.62 + 1.62i)14-s + (−0.0434 + 0.00687i)15-s + (0.325 + 1.00i)16-s + (0.304 − 1.92i)17-s + ⋯
 L(s)  = 1 + (0.494 + 0.160i)2-s + (0.0249 + 0.0249i)3-s + (−0.590 − 0.429i)4-s + (−0.188 + 0.259i)5-s + (0.00833 + 0.0163i)6-s + (−0.537 + 1.05i)7-s + (−0.528 − 0.727i)8-s − 0.998i·9-s + (−0.134 + 0.0980i)10-s + (0.925 + 0.146i)11-s + (−0.00403 − 0.0254i)12-s + (−0.496 + 0.253i)13-s + (−0.434 + 0.434i)14-s + (−0.0112 + 0.00177i)15-s + (0.0813 + 0.250i)16-s + (0.0738 − 0.466i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$41$$ $$\varepsilon$$ = $0.994 - 0.104i$ motivic weight = $$1$$ character : $\chi_{41} (21, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 41,\ (\ :1/2),\ 0.994 - 0.104i)$ $L(1)$ $\approx$ $0.835041 + 0.0436022i$ $L(\frac12)$ $\approx$ $0.835041 + 0.0436022i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 41$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 41$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad41 $$1 + (2.53 + 5.88i)T$$
good2 $$1 + (-0.698 - 0.227i)T + (1.61 + 1.17i)T^{2}$$
3 $$1 + (-0.0432 - 0.0432i)T + 3iT^{2}$$
5 $$1 + (0.422 - 0.581i)T + (-1.54 - 4.75i)T^{2}$$
7 $$1 + (1.42 - 2.79i)T + (-4.11 - 5.66i)T^{2}$$
11 $$1 + (-3.06 - 0.486i)T + (10.4 + 3.39i)T^{2}$$
13 $$1 + (1.79 - 0.912i)T + (7.64 - 10.5i)T^{2}$$
17 $$1 + (-0.304 + 1.92i)T + (-16.1 - 5.25i)T^{2}$$
19 $$1 + (-3.91 - 1.99i)T + (11.1 + 15.3i)T^{2}$$
23 $$1 + (-0.275 + 0.848i)T + (-18.6 - 13.5i)T^{2}$$
29 $$1 + (1.43 + 9.03i)T + (-27.5 + 8.96i)T^{2}$$
31 $$1 + (5.64 - 4.10i)T + (9.57 - 29.4i)T^{2}$$
37 $$1 + (3.49 + 2.53i)T + (11.4 + 35.1i)T^{2}$$
43 $$1 + (10.3 + 3.35i)T + (34.7 + 25.2i)T^{2}$$
47 $$1 + (-5.24 - 10.2i)T + (-27.6 + 38.0i)T^{2}$$
53 $$1 + (-0.465 - 2.94i)T + (-50.4 + 16.3i)T^{2}$$
59 $$1 + (-1.37 + 4.23i)T + (-47.7 - 34.6i)T^{2}$$
61 $$1 + (-11.3 + 3.69i)T + (49.3 - 35.8i)T^{2}$$
67 $$1 + (-8.00 + 1.26i)T + (63.7 - 20.7i)T^{2}$$
71 $$1 + (0.988 + 0.156i)T + (67.5 + 21.9i)T^{2}$$
73 $$1 - 6.49iT - 73T^{2}$$
79 $$1 + (4.43 + 4.43i)T + 79iT^{2}$$
83 $$1 - 7.83T + 83T^{2}$$
89 $$1 + (6.40 - 12.5i)T + (-52.3 - 72.0i)T^{2}$$
97 $$1 + (-6.34 + 1.00i)T + (92.2 - 29.9i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}