# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.849 - 0.528i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (14.0 − 24.3i)5-s + (75.7 + 131. i)7-s + (−138. − 240. i)11-s + (−291. + 505. i)13-s + 1.61e3·17-s + 1.36e3·19-s + (428. − 741. i)23-s + (1.16e3 + 2.02e3i)25-s + (4.26e3 + 7.39e3i)29-s + (−1.46e3 + 2.54e3i)31-s + 4.26e3·35-s + 4.03e3·37-s + (9.44e3 − 1.63e4i)41-s + (1.01e4 + 1.75e4i)43-s + (−147. − 256. i)47-s + ⋯
 L(s)  = 1 + (0.251 − 0.435i)5-s + (0.583 + 1.01i)7-s + (−0.346 − 0.599i)11-s + (−0.479 + 0.829i)13-s + 1.35·17-s + 0.869·19-s + (0.168 − 0.292i)23-s + (0.373 + 0.646i)25-s + (0.942 + 1.63i)29-s + (−0.274 + 0.475i)31-s + 0.587·35-s + 0.484·37-s + (0.877 − 1.52i)41-s + (0.837 + 1.45i)43-s + (−0.00976 − 0.0169i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.849 - 0.528i$ motivic weight = $$5$$ character : $\chi_{108} (37, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.849 - 0.528i)$$ $$L(3)$$ $$\approx$$ $$1.96380 + 0.560988i$$ $$L(\frac12)$$ $$\approx$$ $$1.96380 + 0.560988i$$ $$L(\frac{7}{2})$$ 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$$
3 $$1$$
good5 $$1 + (-14.0 + 24.3i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (-75.7 - 131. i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (138. + 240. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + (291. - 505. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 - 1.61e3T + 1.41e6T^{2}$$
19 $$1 - 1.36e3T + 2.47e6T^{2}$$
23 $$1 + (-428. + 741. i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (-4.26e3 - 7.39e3i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (1.46e3 - 2.54e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 - 4.03e3T + 6.93e7T^{2}$$
41 $$1 + (-9.44e3 + 1.63e4i)T + (-5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (-1.01e4 - 1.75e4i)T + (-7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (147. + 256. i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + 3.03e3T + 4.18e8T^{2}$$
59 $$1 + (8.61e3 - 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (1.28e4 + 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-1.31e4 + 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 7.66e4T + 1.80e9T^{2}$$
73 $$1 - 1.49e3T + 2.07e9T^{2}$$
79 $$1 + (4.96e4 + 8.59e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-2.50e4 - 4.33e4i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + 1.36e5T + 5.58e9T^{2}$$
97 $$1 + (-3.33e4 - 5.77e4i)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}