# Properties

 Label 2-99-33.29-c7-0-16 Degree $2$ Conductor $99$ Sign $-0.552 - 0.833i$ Analytic cond. $30.9261$ Root an. cond. $5.56112$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−17.5 − 12.7i)2-s + (105. + 324. i)4-s + (−159. − 218. i)5-s + (−1.13e3 + 369. i)7-s + (1.43e3 − 4.40e3i)8-s + 5.86e3i·10-s + (4.21e3 + 1.30e3i)11-s + (6.83e3 − 9.40e3i)13-s + (2.46e4 + 8.00e3i)14-s + (−4.57e4 + 3.32e4i)16-s + (−4.81e3 + 3.49e3i)17-s + (2.90e4 + 9.44e3i)19-s + (5.43e4 − 7.47e4i)20-s + (−5.72e4 − 7.66e4i)22-s − 9.65e4i·23-s + ⋯
 L(s)  = 1 + (−1.54 − 1.12i)2-s + (0.824 + 2.53i)4-s + (−0.569 − 0.783i)5-s + (−1.25 + 0.406i)7-s + (0.987 − 3.03i)8-s + 1.85i·10-s + (0.955 + 0.296i)11-s + (0.862 − 1.18i)13-s + (2.39 + 0.779i)14-s + (−2.79 + 2.02i)16-s + (−0.237 + 0.172i)17-s + (0.971 + 0.315i)19-s + (1.51 − 2.09i)20-s + (−1.14 − 1.53i)22-s − 1.65i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$99$$    =    $$3^{2} \cdot 11$$ Sign: $-0.552 - 0.833i$ Analytic conductor: $$30.9261$$ Root analytic conductor: $$5.56112$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{99} (62, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 99,\ (\ :7/2),\ -0.552 - 0.833i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.113205 + 0.210769i$$ $$L(\frac12)$$ $$\approx$$ $$0.113205 + 0.210769i$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1 + (-4.21e3 - 1.30e3i)T$$
good2 $$1 + (17.5 + 12.7i)T + (39.5 + 121. i)T^{2}$$
5 $$1 + (159. + 218. i)T + (-2.41e4 + 7.43e4i)T^{2}$$
7 $$1 + (1.13e3 - 369. i)T + (6.66e5 - 4.84e5i)T^{2}$$
13 $$1 + (-6.83e3 + 9.40e3i)T + (-1.93e7 - 5.96e7i)T^{2}$$
17 $$1 + (4.81e3 - 3.49e3i)T + (1.26e8 - 3.90e8i)T^{2}$$
19 $$1 + (-2.90e4 - 9.44e3i)T + (7.23e8 + 5.25e8i)T^{2}$$
23 $$1 + 9.65e4iT - 3.40e9T^{2}$$
29 $$1 + (2.40e4 + 7.39e4i)T + (-1.39e10 + 1.01e10i)T^{2}$$
31 $$1 + (1.27e5 + 9.24e4i)T + (8.50e9 + 2.61e10i)T^{2}$$
37 $$1 + (-1.35e5 - 4.17e5i)T + (-7.68e10 + 5.57e10i)T^{2}$$
41 $$1 + (7.87e3 - 2.42e4i)T + (-1.57e11 - 1.14e11i)T^{2}$$
43 $$1 - 4.49e4iT - 2.71e11T^{2}$$
47 $$1 + (-2.08e5 - 6.78e4i)T + (4.09e11 + 2.97e11i)T^{2}$$
53 $$1 + (-4.87e5 + 6.70e5i)T + (-3.63e11 - 1.11e12i)T^{2}$$
59 $$1 + (1.99e6 - 6.47e5i)T + (2.01e12 - 1.46e12i)T^{2}$$
61 $$1 + (-5.00e5 - 6.88e5i)T + (-9.71e11 + 2.98e12i)T^{2}$$
67 $$1 + 2.90e6T + 6.06e12T^{2}$$
71 $$1 + (2.16e6 + 2.97e6i)T + (-2.81e12 + 8.64e12i)T^{2}$$
73 $$1 + (-5.27e6 + 1.71e6i)T + (8.93e12 - 6.49e12i)T^{2}$$
79 $$1 + (4.80e6 - 6.61e6i)T + (-5.93e12 - 1.82e13i)T^{2}$$
83 $$1 + (-7.07e5 + 5.14e5i)T + (8.38e12 - 2.58e13i)T^{2}$$
89 $$1 - 3.06e6iT - 4.42e13T^{2}$$
97 $$1 + (2.33e6 + 1.69e6i)T + (2.49e13 + 7.68e13i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$