# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.991 + 0.129i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.253 + 5.65i)2-s + (−31.8 − 2.86i)4-s + (−84.3 + 48.6i)5-s + (−87.9 − 50.7i)7-s + (24.2 − 179. i)8-s + (−253. − 488. i)10-s + (73.2 − 126. i)11-s + (402. + 697. i)13-s + (309. − 484. i)14-s + (1.00e3 + 182. i)16-s + 1.23e3i·17-s − 2.61e3i·19-s + (2.82e3 − 1.30e3i)20-s + (698. + 446. i)22-s + (−84.5 − 146. i)23-s + ⋯
 L(s)  = 1 + (−0.0448 + 0.998i)2-s + (−0.995 − 0.0895i)4-s + (−1.50 + 0.870i)5-s + (−0.678 − 0.391i)7-s + (0.134 − 0.990i)8-s + (−0.802 − 1.54i)10-s + (0.182 − 0.316i)11-s + (0.661 + 1.14i)13-s + (0.421 − 0.660i)14-s + (0.983 + 0.178i)16-s + 1.03i·17-s − 1.66i·19-s + (1.58 − 0.732i)20-s + (0.307 + 0.196i)22-s + (−0.0333 − 0.0577i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.991 + 0.129i$ motivic weight = $$5$$ character : $\chi_{108} (35, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.991 + 0.129i)$$ $$L(3)$$ $$\approx$$ $$0.574971 - 0.0373996i$$ $$L(\frac12)$$ $$\approx$$ $$0.574971 - 0.0373996i$$ $$L(\frac{7}{2})$$ 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 + (0.253 - 5.65i)T$$
3 $$1$$
good5 $$1 + (84.3 - 48.6i)T + (1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (87.9 + 50.7i)T + (8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (-73.2 + 126. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (-402. - 697. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 - 1.23e3iT - 1.41e6T^{2}$$
19 $$1 + 2.61e3iT - 2.47e6T^{2}$$
23 $$1 + (84.5 + 146. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (1.95e3 + 1.12e3i)T + (1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-1.78e3 + 1.03e3i)T + (1.43e7 - 2.47e7i)T^{2}$$
37 $$1 - 445.T + 6.93e7T^{2}$$
41 $$1 + (-1.72e3 + 993. i)T + (5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (-3.46e3 - 2.00e3i)T + (7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (-3.53e3 + 6.13e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 - 6.23e3iT - 4.18e8T^{2}$$
59 $$1 + (1.67e4 + 2.89e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-1.47e4 + 2.54e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-2.33e4 + 1.34e4i)T + (6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 3.63e4T + 1.80e9T^{2}$$
73 $$1 + 3.82e4T + 2.07e9T^{2}$$
79 $$1 + (-5.45e4 - 3.14e4i)T + (1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-3.99e4 + 6.92e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + 9.66e4iT - 5.58e9T^{2}$$
97 $$1 + (8.31e3 - 1.44e4i)T + (-4.29e9 - 7.43e9i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}