# Properties

 Label 2-91-91.4-c1-0-5 Degree $2$ Conductor $91$ Sign $-0.770 + 0.637i$ 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 − 1.37i·2-s + (−1.44 − 2.49i)3-s + 0.0982·4-s + (−0.697 + 0.402i)5-s + (−3.44 + 1.98i)6-s + (0.0699 + 2.64i)7-s − 2.89i·8-s + (−2.65 + 4.59i)9-s + (0.555 + 0.962i)10-s + (4.56 − 2.63i)11-s + (−0.141 − 0.245i)12-s + (−2.36 − 2.72i)13-s + (3.64 − 0.0965i)14-s + (2.01 + 1.16i)15-s − 3.79·16-s + 0.560·17-s + ⋯
 L(s)  = 1 − 0.975i·2-s + (−0.831 − 1.44i)3-s + 0.0491·4-s + (−0.312 + 0.180i)5-s + (−1.40 + 0.811i)6-s + (0.0264 + 0.999i)7-s − 1.02i·8-s + (−0.883 + 1.53i)9-s + (0.175 + 0.304i)10-s + (1.37 − 0.794i)11-s + (−0.0408 − 0.0707i)12-s + (−0.656 − 0.754i)13-s + (0.974 − 0.0257i)14-s + (0.519 + 0.299i)15-s − 0.948·16-s + 0.135·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $-0.770 + 0.637i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (4, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ -0.770 + 0.637i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.283574 - 0.787081i$$ $$L(\frac12)$$ $$\approx$$ $$0.283574 - 0.787081i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-0.0699 - 2.64i)T$$
13 $$1 + (2.36 + 2.72i)T$$
good2 $$1 + 1.37iT - 2T^{2}$$
3 $$1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (0.697 - 0.402i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-4.56 + 2.63i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 - 0.560T + 17T^{2}$$
19 $$1 + (-5.06 - 2.92i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 - 1.60T + 23T^{2}$$
29 $$1 + (1.14 - 1.97i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-3.01 - 1.73i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 1.24iT - 37T^{2}$$
41 $$1 + (-0.803 - 0.463i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (3.32 - 1.92i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (2.72 - 4.72i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + 10.9iT - 59T^{2}$$
61 $$1 + (3.65 - 6.32i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-6.36 + 3.67i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (8.06 - 4.65i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-4.33 - 2.50i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (5.68 + 9.84i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 5.81iT - 83T^{2}$$
89 $$1 - 5.00iT - 89T^{2}$$
97 $$1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$