# Properties

 Label 2-39e2-13.12-c1-0-16 Degree $2$ Conductor $1521$ Sign $-0.691 - 0.722i$ Analytic cond. $12.1452$ Root an. cond. $3.48500$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.69i·2-s − 5.24·4-s − 1.04i·5-s + 0.554i·7-s − 8.74i·8-s + 2.82·10-s − 2.91i·11-s − 1.49·14-s + 13.0·16-s − 4.85·17-s + 0.753i·19-s + 5.50i·20-s + 7.83·22-s + 5.76·23-s + 3.89·25-s + ⋯
 L(s)  = 1 + 1.90i·2-s − 2.62·4-s − 0.469i·5-s + 0.209i·7-s − 3.09i·8-s + 0.892·10-s − 0.877i·11-s − 0.399·14-s + 3.25·16-s − 1.17·17-s + 0.172i·19-s + 1.23i·20-s + 1.67·22-s + 1.20·23-s + 0.779·25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1521$$    =    $$3^{2} \cdot 13^{2}$$ Sign: $-0.691 - 0.722i$ Analytic conductor: $$12.1452$$ Root analytic conductor: $$3.48500$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1521} (1351, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1521,\ (\ :1/2),\ -0.691 - 0.722i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.275052131$$ $$L(\frac12)$$ $$\approx$$ $$1.275052131$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
13 $$1$$
good2 $$1 - 2.69iT - 2T^{2}$$
5 $$1 + 1.04iT - 5T^{2}$$
7 $$1 - 0.554iT - 7T^{2}$$
11 $$1 + 2.91iT - 11T^{2}$$
17 $$1 + 4.85T + 17T^{2}$$
19 $$1 - 0.753iT - 19T^{2}$$
23 $$1 - 5.76T + 23T^{2}$$
29 $$1 - 1.91T + 29T^{2}$$
31 $$1 - 9.51iT - 31T^{2}$$
37 $$1 - 5.75iT - 37T^{2}$$
41 $$1 - 4.91iT - 41T^{2}$$
43 $$1 - 11.0T + 43T^{2}$$
47 $$1 + 0.753iT - 47T^{2}$$
53 $$1 - 7.58T + 53T^{2}$$
59 $$1 + 4.09iT - 59T^{2}$$
61 $$1 + 3.42T + 61T^{2}$$
67 $$1 - 1.87iT - 67T^{2}$$
71 $$1 + 10.5iT - 71T^{2}$$
73 $$1 - 10.4iT - 73T^{2}$$
79 $$1 - 1.33T + 79T^{2}$$
83 $$1 - 2.64iT - 83T^{2}$$
89 $$1 + 9.92iT - 89T^{2}$$
97 $$1 + 17.0iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$