# Properties

 Label 2-91-91.6-c1-0-7 Degree $2$ Conductor $91$ Sign $-0.811 - 0.584i$ 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 + (−0.607 − 2.26i)2-s + (−2.39 − 1.38i)3-s + (−3.03 + 1.75i)4-s + (1.12 + 1.12i)5-s + (−1.67 + 6.25i)6-s + (−0.437 − 2.60i)7-s + (2.50 + 2.50i)8-s + (2.30 + 4.00i)9-s + (1.87 − 3.23i)10-s + (−3.03 + 0.813i)11-s + 9.68·12-s + (−1.04 − 3.44i)13-s + (−5.64 + 2.57i)14-s + (−1.13 − 4.24i)15-s + (0.649 − 1.12i)16-s + (0.320 + 0.555i)17-s + ⋯
 L(s)  = 1 + (−0.429 − 1.60i)2-s + (−1.38 − 0.796i)3-s + (−1.51 + 0.877i)4-s + (0.503 + 0.503i)5-s + (−0.684 + 2.55i)6-s + (−0.165 − 0.986i)7-s + (0.886 + 0.886i)8-s + (0.769 + 1.33i)9-s + (0.591 − 1.02i)10-s + (−0.914 + 0.245i)11-s + 2.79·12-s + (−0.290 − 0.956i)13-s + (−1.51 + 0.689i)14-s + (−0.293 − 1.09i)15-s + (0.162 − 0.281i)16-s + (0.0778 + 0.134i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $-0.811 - 0.584i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ -0.811 - 0.584i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.128482 + 0.398018i$$ $$L(\frac12)$$ $$\approx$$ $$0.128482 + 0.398018i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (0.437 + 2.60i)T$$
13 $$1 + (1.04 + 3.44i)T$$
good2 $$1 + (0.607 + 2.26i)T + (-1.73 + i)T^{2}$$
3 $$1 + (2.39 + 1.38i)T + (1.5 + 2.59i)T^{2}$$
5 $$1 + (-1.12 - 1.12i)T + 5iT^{2}$$
11 $$1 + (3.03 - 0.813i)T + (9.52 - 5.5i)T^{2}$$
17 $$1 + (-0.320 - 0.555i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.04 + 7.61i)T + (-16.4 - 9.5i)T^{2}$$
23 $$1 + (0.126 + 0.0730i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-4.73 - 4.73i)T + 31iT^{2}$$
37 $$1 + (-3.75 + 1.00i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 + (-5.60 + 1.50i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (2.22 - 2.22i)T - 47iT^{2}$$
53 $$1 + 7.32T + 53T^{2}$$
59 $$1 + (4.00 + 1.07i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (3.90 - 2.25i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (-13.8 - 3.70i)T + (61.4 + 35.5i)T^{2}$$
73 $$1 + (-4.99 + 4.99i)T - 73iT^{2}$$
79 $$1 + 0.632T + 79T^{2}$$
83 $$1 + (-1.07 - 1.07i)T + 83iT^{2}$$
89 $$1 + (-3.51 - 13.1i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 + (-0.0487 + 0.181i)T + (-84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$