Properties

 Degree 2 Conductor $7 \cdot 571$ Sign $0.893 - 0.449i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.341 − 1.74i)2-s + (−2.00 − 0.815i)4-s + (−0.461 + 0.887i)7-s + (−1.13 + 1.73i)8-s + (−0.739 − 0.673i)9-s + (−0.000667 + 0.121i)11-s + (1.39 + 1.10i)14-s + (1.09 + 1.06i)16-s + (−1.42 + 1.06i)18-s + (0.211 + 0.0425i)22-s + (−1.65 + 0.448i)23-s + (−0.795 − 0.605i)25-s + (1.64 − 1.40i)28-s + (−0.0415 + 1.50i)29-s + (0.527 − 0.366i)32-s + ⋯
 L(s)  = 1 + (0.341 − 1.74i)2-s + (−2.00 − 0.815i)4-s + (−0.461 + 0.887i)7-s + (−1.13 + 1.73i)8-s + (−0.739 − 0.673i)9-s + (−0.000667 + 0.121i)11-s + (1.39 + 1.10i)14-s + (1.09 + 1.06i)16-s + (−1.42 + 1.06i)18-s + (0.211 + 0.0425i)22-s + (−1.65 + 0.448i)23-s + (−0.795 − 0.605i)25-s + (1.64 − 1.40i)28-s + (−0.0415 + 1.50i)29-s + (0.527 − 0.366i)32-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$3997$$    =    $$7 \cdot 571$$ $$\varepsilon$$ = $0.893 - 0.449i$ motivic weight = $$0$$ character : $\chi_{3997} (13, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3997,\ (\ :0),\ 0.893 - 0.449i)$ $L(\frac{1}{2})$ $\approx$ $0.2626400568$ $L(\frac12)$ $\approx$ $0.2626400568$ $L(1)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{7,\;571\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{7,\;571\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 $$1 + (0.461 - 0.887i)T$$
571 $$1 + (-0.716 - 0.697i)T$$
good2 $$1 + (-0.341 + 1.74i)T + (-0.926 - 0.376i)T^{2}$$
3 $$1 + (0.739 + 0.673i)T^{2}$$
5 $$1 + (0.795 + 0.605i)T^{2}$$
11 $$1 + (0.000667 - 0.121i)T + (-0.999 - 0.0110i)T^{2}$$
13 $$1 + (0.709 - 0.705i)T^{2}$$
17 $$1 + (-0.959 - 0.282i)T^{2}$$
19 $$1 + (-0.202 - 0.979i)T^{2}$$
23 $$1 + (1.65 - 0.448i)T + (0.863 - 0.504i)T^{2}$$
29 $$1 + (0.0415 - 1.50i)T + (-0.998 - 0.0550i)T^{2}$$
31 $$1 + (0.879 + 0.475i)T^{2}$$
37 $$1 + (0.977 + 0.0756i)T + (0.988 + 0.153i)T^{2}$$
41 $$1 + (0.821 - 0.569i)T^{2}$$
43 $$1 + (-0.719 - 1.59i)T + (-0.660 + 0.750i)T^{2}$$
47 $$1 + (-0.851 + 0.523i)T^{2}$$
53 $$1 + (-1.39 - 1.29i)T + (0.0715 + 0.997i)T^{2}$$
59 $$1 + (-0.945 - 0.324i)T^{2}$$
61 $$1 + (-0.471 + 0.882i)T^{2}$$
67 $$1 + (0.0859 - 0.373i)T + (-0.899 - 0.436i)T^{2}$$
71 $$1 + (1.46 - 0.310i)T + (0.913 - 0.406i)T^{2}$$
73 $$1 + (0.360 + 0.932i)T^{2}$$
79 $$1 + (1.77 + 0.863i)T + (0.618 + 0.785i)T^{2}$$
83 $$1 + (0.997 + 0.0770i)T^{2}$$
89 $$1 + (-0.565 + 0.824i)T^{2}$$
97 $$1 + (0.441 - 0.897i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}