# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.891 + 0.453i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.0634 − 2.99i)3-s + (5.64 + 6.73i)5-s + (4.05 − 1.47i)7-s + (−8.99 − 0.380i)9-s + (12.6 − 15.0i)11-s + (−1.31 − 7.45i)13-s + (20.5 − 16.5i)15-s + (−3.16 + 1.82i)17-s + (−16.8 + 29.1i)19-s + (−4.16 − 12.2i)21-s + (1.05 − 2.88i)23-s + (−9.07 + 51.4i)25-s + (−1.71 + 26.9i)27-s + (23.9 + 4.22i)29-s + (−27.1 − 9.87i)31-s + ⋯
 L(s)  = 1 + (0.0211 − 0.999i)3-s + (1.12 + 1.34i)5-s + (0.578 − 0.210i)7-s + (−0.999 − 0.0422i)9-s + (1.15 − 1.37i)11-s + (−0.101 − 0.573i)13-s + (1.37 − 1.10i)15-s + (−0.186 + 0.107i)17-s + (−0.885 + 1.53i)19-s + (−0.198 − 0.583i)21-s + (0.0457 − 0.125i)23-s + (−0.362 + 2.05i)25-s + (−0.0634 + 0.997i)27-s + (0.825 + 0.145i)29-s + (−0.875 − 0.318i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.891 + 0.453i$ motivic weight = $$2$$ character : $\chi_{108} (29, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :1),\ 0.891 + 0.453i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.59417 - 0.382276i$$ $$L(\frac12)$$ $$\approx$$ $$1.59417 - 0.382276i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-0.0634 + 2.99i)T$$
good5 $$1 + (-5.64 - 6.73i)T + (-4.34 + 24.6i)T^{2}$$
7 $$1 + (-4.05 + 1.47i)T + (37.5 - 31.4i)T^{2}$$
11 $$1 + (-12.6 + 15.0i)T + (-21.0 - 119. i)T^{2}$$
13 $$1 + (1.31 + 7.45i)T + (-158. + 57.8i)T^{2}$$
17 $$1 + (3.16 - 1.82i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (16.8 - 29.1i)T + (-180.5 - 312. i)T^{2}$$
23 $$1 + (-1.05 + 2.88i)T + (-405. - 340. i)T^{2}$$
29 $$1 + (-23.9 - 4.22i)T + (790. + 287. i)T^{2}$$
31 $$1 + (27.1 + 9.87i)T + (736. + 617. i)T^{2}$$
37 $$1 + (14.9 + 25.8i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + (22.6 - 3.99i)T + (1.57e3 - 574. i)T^{2}$$
43 $$1 + (6.85 + 5.74i)T + (321. + 1.82e3i)T^{2}$$
47 $$1 + (-3.60 - 9.90i)T + (-1.69e3 + 1.41e3i)T^{2}$$
53 $$1 - 70.8iT - 2.80e3T^{2}$$
59 $$1 + (43.8 + 52.3i)T + (-604. + 3.42e3i)T^{2}$$
61 $$1 + (99.7 - 36.3i)T + (2.85e3 - 2.39e3i)T^{2}$$
67 $$1 + (4.27 + 24.2i)T + (-4.21e3 + 1.53e3i)T^{2}$$
71 $$1 + (-29.9 + 17.2i)T + (2.52e3 - 4.36e3i)T^{2}$$
73 $$1 + (-20.9 + 36.2i)T + (-2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (-8.78 + 49.8i)T + (-5.86e3 - 2.13e3i)T^{2}$$
83 $$1 + (30.9 + 5.45i)T + (6.47e3 + 2.35e3i)T^{2}$$
89 $$1 + (-40.0 - 23.1i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + (-84.8 - 71.1i)T + (1.63e3 + 9.26e3i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}