# Properties

 Label 2-13-13.12-c3-0-1 Degree $2$ Conductor $13$ Sign $0.554 + 0.832i$ Analytic cond. $0.767024$ Root an. cond. $0.875799$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·2-s − 3-s − 4-s + 9i·5-s + 3i·6-s + 15i·7-s − 21i·8-s − 26·9-s + 27·10-s − 48i·11-s + 12-s + (26 + 39i)13-s + 45·14-s − 9i·15-s − 71·16-s − 45·17-s + ⋯
 L(s)  = 1 − 1.06i·2-s − 0.192·3-s − 0.125·4-s + 0.804i·5-s + 0.204i·6-s + 0.809i·7-s − 0.928i·8-s − 0.962·9-s + 0.853·10-s − 1.31i·11-s + 0.0240·12-s + (0.554 + 0.832i)13-s + 0.859·14-s − 0.154i·15-s − 1.10·16-s − 0.642·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$13$$ Sign: $0.554 + 0.832i$ Analytic conductor: $$0.767024$$ Root analytic conductor: $$0.875799$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{13} (12, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 13,\ (\ :3/2),\ 0.554 + 0.832i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.825202 - 0.441634i$$ $$L(\frac12)$$ $$\approx$$ $$0.825202 - 0.441634i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1 + (-26 - 39i)T$$
good2 $$1 + 3iT - 8T^{2}$$
3 $$1 + T + 27T^{2}$$
5 $$1 - 9iT - 125T^{2}$$
7 $$1 - 15iT - 343T^{2}$$
11 $$1 + 48iT - 1.33e3T^{2}$$
17 $$1 + 45T + 4.91e3T^{2}$$
19 $$1 + 6iT - 6.85e3T^{2}$$
23 $$1 - 162T + 1.21e4T^{2}$$
29 $$1 + 144T + 2.43e4T^{2}$$
31 $$1 + 264iT - 2.97e4T^{2}$$
37 $$1 - 303iT - 5.06e4T^{2}$$
41 $$1 - 192iT - 6.89e4T^{2}$$
43 $$1 + 97T + 7.95e4T^{2}$$
47 $$1 - 111iT - 1.03e5T^{2}$$
53 $$1 + 414T + 1.48e5T^{2}$$
59 $$1 - 522iT - 2.05e5T^{2}$$
61 $$1 - 376T + 2.26e5T^{2}$$
67 $$1 - 36iT - 3.00e5T^{2}$$
71 $$1 + 357iT - 3.57e5T^{2}$$
73 $$1 + 1.09e3iT - 3.89e5T^{2}$$
79 $$1 + 830T + 4.93e5T^{2}$$
83 $$1 - 438iT - 5.71e5T^{2}$$
89 $$1 + 438iT - 7.04e5T^{2}$$
97 $$1 - 852iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$