Properties

 Label 2-507-13.12-c3-0-57 Degree $2$ Conductor $507$ Sign $0.832 + 0.554i$ Analytic cond. $29.9139$ Root an. cond. $5.46936$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + i·2-s + 3·3-s + 7·4-s − 7i·5-s + 3i·6-s − 10i·7-s + 15i·8-s + 9·9-s + 7·10-s − 22i·11-s + 21·12-s + 10·14-s − 21i·15-s + 41·16-s − 37·17-s + 9i·18-s + ⋯
 L(s)  = 1 + 0.353i·2-s + 0.577·3-s + 0.875·4-s − 0.626i·5-s + 0.204i·6-s − 0.539i·7-s + 0.662i·8-s + 0.333·9-s + 0.221·10-s − 0.603i·11-s + 0.505·12-s + 0.190·14-s − 0.361i·15-s + 0.640·16-s − 0.527·17-s + 0.117i·18-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $0.832 + 0.554i$ Analytic conductor: $$29.9139$$ Root analytic conductor: $$5.46936$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{507} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :3/2),\ 0.832 + 0.554i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$3.073958774$$ $$L(\frac12)$$ $$\approx$$ $$3.073958774$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
13 $$1$$
good2 $$1 - iT - 8T^{2}$$
5 $$1 + 7iT - 125T^{2}$$
7 $$1 + 10iT - 343T^{2}$$
11 $$1 + 22iT - 1.33e3T^{2}$$
17 $$1 + 37T + 4.91e3T^{2}$$
19 $$1 + 30iT - 6.85e3T^{2}$$
23 $$1 - 162T + 1.21e4T^{2}$$
29 $$1 + 113T + 2.43e4T^{2}$$
31 $$1 + 196iT - 2.97e4T^{2}$$
37 $$1 - 13iT - 5.06e4T^{2}$$
41 $$1 + 285iT - 6.89e4T^{2}$$
43 $$1 - 246T + 7.95e4T^{2}$$
47 $$1 + 462iT - 1.03e5T^{2}$$
53 $$1 + 537T + 1.48e5T^{2}$$
59 $$1 - 576iT - 2.05e5T^{2}$$
61 $$1 + 635T + 2.26e5T^{2}$$
67 $$1 + 202iT - 3.00e5T^{2}$$
71 $$1 - 1.08e3iT - 3.57e5T^{2}$$
73 $$1 + 805iT - 3.89e5T^{2}$$
79 $$1 - 884T + 4.93e5T^{2}$$
83 $$1 + 518iT - 5.71e5T^{2}$$
89 $$1 - 194iT - 7.04e5T^{2}$$
97 $$1 - 1.20e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$