Properties

 Degree 2 Conductor $2^{5} \cdot 5^{3}$ Sign $0.936 - 0.349i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−0.587 + 0.809i)9-s + (−0.142 − 0.896i)13-s + (0.278 + 0.142i)17-s + (1.11 + 0.363i)29-s + (1.95 − 0.309i)37-s + (1.53 + 1.11i)41-s + i·49-s + (1.58 − 0.809i)53-s + (−1.53 + 1.11i)61-s + (−1.76 − 0.278i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.896 + 1.76i)97-s + 1.61·101-s + (0.363 − 0.5i)109-s + ⋯
 L(s)  = 1 + (−0.587 + 0.809i)9-s + (−0.142 − 0.896i)13-s + (0.278 + 0.142i)17-s + (1.11 + 0.363i)29-s + (1.95 − 0.309i)37-s + (1.53 + 1.11i)41-s + i·49-s + (1.58 − 0.809i)53-s + (−1.53 + 1.11i)61-s + (−1.76 − 0.278i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.896 + 1.76i)97-s + 1.61·101-s + (0.363 − 0.5i)109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$4000$$    =    $$2^{5} \cdot 5^{3}$$ $$\varepsilon$$ = $0.936 - 0.349i$ motivic weight = $$0$$ character : $\chi_{4000} (2593, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4000,\ (\ :0),\ 0.936 - 0.349i)$ $L(\frac{1}{2})$ $\approx$ $1.243683497$ $L(\frac12)$ $\approx$ $1.243683497$ $L(1)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + (0.587 - 0.809i)T^{2}$$
7 $$1 - iT^{2}$$
11 $$1 + (0.309 - 0.951i)T^{2}$$
13 $$1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2}$$
17 $$1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2}$$
19 $$1 + (0.809 - 0.587i)T^{2}$$
23 $$1 + (0.951 + 0.309i)T^{2}$$
29 $$1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2}$$
31 $$1 + (-0.809 + 0.587i)T^{2}$$
37 $$1 + (-1.95 + 0.309i)T + (0.951 - 0.309i)T^{2}$$
41 $$1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + (-0.587 + 0.809i)T^{2}$$
53 $$1 + (-1.58 + 0.809i)T + (0.587 - 0.809i)T^{2}$$
59 $$1 + (-0.309 - 0.951i)T^{2}$$
61 $$1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2}$$
67 $$1 + (0.587 + 0.809i)T^{2}$$
71 $$1 + (-0.809 - 0.587i)T^{2}$$
73 $$1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2}$$
79 $$1 + (0.809 + 0.587i)T^{2}$$
83 $$1 + (-0.587 - 0.809i)T^{2}$$
89 $$1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2}$$
97 $$1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}