Properties

 Label 2-39e2-13.8-c0-0-2 Degree $2$ Conductor $1521$ Sign $0.289 + 0.957i$ Analytic cond. $0.759077$ Root an. cond. $0.871250$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − i·4-s + (1 − i)7-s − 16-s + (1 + i)19-s − i·25-s + (−1 − i)28-s + (−1 − i)31-s + (−1 + i)37-s − i·49-s + i·64-s + (1 + i)67-s + (1 − i)73-s + (1 − i)76-s + (−1 − i)97-s − 100-s + ⋯
 L(s)  = 1 − i·4-s + (1 − i)7-s − 16-s + (1 + i)19-s − i·25-s + (−1 − i)28-s + (−1 − i)31-s + (−1 + i)37-s − i·49-s + i·64-s + (1 + i)67-s + (1 − i)73-s + (1 − i)76-s + (−1 − i)97-s − 100-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1521$$    =    $$3^{2} \cdot 13^{2}$$ Sign: $0.289 + 0.957i$ Analytic conductor: $$0.759077$$ Root analytic conductor: $$0.871250$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1521} (775, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1521,\ (\ :0),\ 0.289 + 0.957i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.201909156$$ $$L(\frac12)$$ $$\approx$$ $$1.201909156$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
13 $$1$$
good2 $$1 + iT^{2}$$
5 $$1 + iT^{2}$$
7 $$1 + (-1 + i)T - iT^{2}$$
11 $$1 - iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + (-1 - i)T + iT^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (1 + i)T + iT^{2}$$
37 $$1 + (1 - i)T - iT^{2}$$
41 $$1 + iT^{2}$$
43 $$1 - T^{2}$$
47 $$1 - iT^{2}$$
53 $$1 + T^{2}$$
59 $$1 - iT^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (-1 - i)T + iT^{2}$$
71 $$1 + iT^{2}$$
73 $$1 + (-1 + i)T - iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 - iT^{2}$$
97 $$1 + (1 + i)T + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$