# Properties

 Degree 2 Conductor $3^{2} \cdot 11$ Sign $-0.357 - 0.934i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.23 + 2.13i)2-s + (1.66 − 0.461i)3-s + (−2.04 − 3.54i)4-s + (1.21 + 2.10i)5-s + (−1.07 + 4.14i)6-s + (−1.16 + 2.02i)7-s + 5.17·8-s + (2.57 − 1.54i)9-s − 6.01·10-s + (−0.5 + 0.866i)11-s + (−5.05 − 4.97i)12-s + (−2.35 − 4.07i)13-s + (−2.88 − 5.00i)14-s + (3.00 + 2.95i)15-s + (−2.29 + 3.97i)16-s − 3.20·17-s + ⋯
 L(s)  = 1 + (−0.873 + 1.51i)2-s + (0.963 − 0.266i)3-s + (−1.02 − 1.77i)4-s + (0.544 + 0.943i)5-s + (−0.438 + 1.69i)6-s + (−0.441 + 0.765i)7-s + 1.83·8-s + (0.857 − 0.514i)9-s − 1.90·10-s + (−0.150 + 0.261i)11-s + (−1.46 − 1.43i)12-s + (−0.653 − 1.13i)13-s + (−0.771 − 1.33i)14-s + (0.776 + 0.764i)15-s + (−0.574 + 0.994i)16-s − 0.778·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$99$$    =    $$3^{2} \cdot 11$$ $$\varepsilon$$ = $-0.357 - 0.934i$ motivic weight = $$1$$ character : $\chi_{99} (34, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 99,\ (\ :1/2),\ -0.357 - 0.934i)$$ $$L(1)$$ $$\approx$$ $$0.490556 + 0.712859i$$ $$L(\frac12)$$ $$\approx$$ $$0.490556 + 0.712859i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;11\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-1.66 + 0.461i)T$$
11 $$1 + (0.5 - 0.866i)T$$
good2 $$1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (-1.21 - 2.10i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2}$$
13 $$1 + (2.35 + 4.07i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + 3.20T + 17T^{2}$$
19 $$1 - 7.77T + 19T^{2}$$
23 $$1 + (1.37 + 2.38i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.18 + 2.05i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-0.685 - 1.18i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 8.47T + 37T^{2}$$
41 $$1 + (1.77 + 3.07i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (0.103 - 0.180i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 9.11T + 53T^{2}$$
59 $$1 + (-0.120 - 0.208i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (0.830 - 1.43i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 1.07T + 71T^{2}$$
73 $$1 + 2.37T + 73T^{2}$$
79 $$1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (5.25 - 9.09i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + 14.2T + 89T^{2}$$
97 $$1 + (4.46 - 7.73i)T + (-48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}