# Properties

 Label 2-91-91.16-c1-0-6 Degree $2$ Conductor $91$ Sign $0.788 + 0.615i$ Analytic cond. $0.726638$ Root an. cond. $0.852431$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (1.5 − 2.59i)3-s − 4-s + (−1.5 + 2.59i)5-s + (1.5 − 2.59i)6-s + (2 + 1.73i)7-s − 3·8-s + (−3 − 5.19i)9-s + (−1.5 + 2.59i)10-s + (1.5 − 2.59i)11-s + (−1.5 + 2.59i)12-s + (−1 + 3.46i)13-s + (2 + 1.73i)14-s + (4.5 + 7.79i)15-s − 16-s − 2·17-s + ⋯
 L(s)  = 1 + 0.707·2-s + (0.866 − 1.49i)3-s − 0.5·4-s + (−0.670 + 1.16i)5-s + (0.612 − 1.06i)6-s + (0.755 + 0.654i)7-s − 1.06·8-s + (−1 − 1.73i)9-s + (−0.474 + 0.821i)10-s + (0.452 − 0.783i)11-s + (−0.433 + 0.749i)12-s + (−0.277 + 0.960i)13-s + (0.534 + 0.462i)14-s + (1.16 + 2.01i)15-s − 0.250·16-s − 0.485·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $0.788 + 0.615i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (16, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ 0.788 + 0.615i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.31154 - 0.451563i$$ $$L(\frac12)$$ $$\approx$$ $$1.31154 - 0.451563i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-2 - 1.73i)T$$
13 $$1 + (1 - 3.46i)T$$
good2 $$1 - T + 2T^{2}$$
3 $$1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + 2T + 17T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$