Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 7$ Sign $0.920 + 0.391i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−2 − i)5-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (0.707 + 2.12i)10-s + 0.585i·11-s + (4 + 4i)13-s + 1.00·14-s − 1.00·16-s + (0.585 + 0.585i)17-s − 2.82i·19-s + (1.00 − 2.00i)20-s + (0.414 − 0.414i)22-s + (4.82 − 4.82i)23-s + ⋯
 L(s)  = 1 + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.894 − 0.447i)5-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (0.223 + 0.670i)10-s + 0.176i·11-s + (1.10 + 1.10i)13-s + 0.267·14-s − 0.250·16-s + (0.142 + 0.142i)17-s − 0.648i·19-s + (0.223 − 0.447i)20-s + (0.0883 − 0.0883i)22-s + (1.00 − 1.00i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$630$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.920 + 0.391i$ motivic weight = $$1$$ character : $\chi_{630} (323, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 630,\ (\ :1/2),\ 0.920 + 0.391i)$$ $$L(1)$$ $$\approx$$ $$0.948715 - 0.193174i$$ $$L(\frac12)$$ $$\approx$$ $$0.948715 - 0.193174i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.707 + 0.707i)T$$
3 $$1$$
5 $$1 + (2 + i)T$$
7 $$1 + (0.707 - 0.707i)T$$
good11 $$1 - 0.585iT - 11T^{2}$$
13 $$1 + (-4 - 4i)T + 13iT^{2}$$
17 $$1 + (-0.585 - 0.585i)T + 17iT^{2}$$
19 $$1 + 2.82iT - 19T^{2}$$
23 $$1 + (-4.82 + 4.82i)T - 23iT^{2}$$
29 $$1 - 0.828T + 29T^{2}$$
31 $$1 - 1.75T + 31T^{2}$$
37 $$1 + (-6.24 + 6.24i)T - 37iT^{2}$$
41 $$1 + 3.17iT - 41T^{2}$$
43 $$1 + (-6.07 - 6.07i)T + 43iT^{2}$$
47 $$1 + (-9.24 - 9.24i)T + 47iT^{2}$$
53 $$1 + (2.58 - 2.58i)T - 53iT^{2}$$
59 $$1 - 2.82T + 59T^{2}$$
61 $$1 + 9.89T + 61T^{2}$$
67 $$1 + (-8.41 + 8.41i)T - 67iT^{2}$$
71 $$1 - 1.17iT - 71T^{2}$$
73 $$1 + (-7.07 - 7.07i)T + 73iT^{2}$$
79 $$1 + 5.65iT - 79T^{2}$$
83 $$1 + (0.828 - 0.828i)T - 83iT^{2}$$
89 $$1 - 3.17T + 89T^{2}$$
97 $$1 + (4.58 - 4.58i)T - 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}