Properties

 Label 2-91-91.4-c1-0-4 Degree $2$ Conductor $91$ Sign $0.372 + 0.927i$ 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 − 1.73i·2-s + (0.5 + 0.866i)3-s − 0.999·4-s + (1.5 − 0.866i)5-s + (1.49 − 0.866i)6-s + (−2 + 1.73i)7-s − 1.73i·8-s + (1 − 1.73i)9-s + (−1.49 − 2.59i)10-s + (−4.5 + 2.59i)11-s + (−0.499 − 0.866i)12-s + (−1 + 3.46i)13-s + (2.99 + 3.46i)14-s + (1.5 + 0.866i)15-s − 5·16-s + 6·17-s + ⋯
 L(s)  = 1 − 1.22i·2-s + (0.288 + 0.499i)3-s − 0.499·4-s + (0.670 − 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.755 + 0.654i)7-s − 0.612i·8-s + (0.333 − 0.577i)9-s + (−0.474 − 0.821i)10-s + (−1.35 + 0.783i)11-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.960i)13-s + (0.801 + 0.925i)14-s + (0.387 + 0.223i)15-s − 1.25·16-s + 1.45·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $0.372 + 0.927i$ 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: $$(2,\ 91,\ (\ :1/2),\ 0.372 + 0.927i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.921335 - 0.622678i$$ $$L(\frac12)$$ $$\approx$$ $$0.921335 - 0.622678i$$ $$L(\frac{3}{2})$$ 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 + 1.73iT - 2T^{2}$$
3 $$1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 - 6T + 17T^{2}$$
19 $$1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-7.5 + 4.33i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 3.46iT - 59T^{2}$$
61 $$1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 3.46iT - 83T^{2}$$
89 $$1 + 6.92iT - 89T^{2}$$
97 $$1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$