# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.590 + 0.807i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−13.1 + 22.7i)5-s + (−31.6 − 54.7i)7-s + (49.1 + 85.0i)11-s + (369. − 639. i)13-s − 250.·17-s + 1.10e3·19-s + (2.20e3 − 3.81e3i)23-s + (1.21e3 + 2.10e3i)25-s + (−3.94e3 − 6.82e3i)29-s + (2.30e3 − 3.99e3i)31-s + 1.66e3·35-s + 1.18e4·37-s + (5.04e3 − 8.73e3i)41-s + (−3.51e3 − 6.09e3i)43-s + (7.45e3 + 1.29e4i)47-s + ⋯
 L(s)  = 1 + (−0.235 + 0.407i)5-s + (−0.243 − 0.422i)7-s + (0.122 + 0.211i)11-s + (0.605 − 1.04i)13-s − 0.209·17-s + 0.700·19-s + (0.869 − 1.50i)23-s + (0.389 + 0.674i)25-s + (−0.870 − 1.50i)29-s + (0.430 − 0.746i)31-s + 0.229·35-s + 1.42·37-s + (0.468 − 0.811i)41-s + (−0.290 − 0.502i)43-s + (0.492 + 0.853i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.590 + 0.807i$ motivic weight = $$5$$ character : $\chi_{108} (37, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.590 + 0.807i)$$ $$L(3)$$ $$\approx$$ $$1.42570 - 0.723620i$$ $$L(\frac12)$$ $$\approx$$ $$1.42570 - 0.723620i$$ $$L(\frac{7}{2})$$ 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 + (13.1 - 22.7i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (31.6 + 54.7i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (-49.1 - 85.0i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + (-369. + 639. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + 250.T + 1.41e6T^{2}$$
19 $$1 - 1.10e3T + 2.47e6T^{2}$$
23 $$1 + (-2.20e3 + 3.81e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (3.94e3 + 6.82e3i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-2.30e3 + 3.99e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 - 1.18e4T + 6.93e7T^{2}$$
41 $$1 + (-5.04e3 + 8.73e3i)T + (-5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (3.51e3 + 6.09e3i)T + (-7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (-7.45e3 - 1.29e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + 2.24e4T + 4.18e8T^{2}$$
59 $$1 + (-5.40e3 + 9.36e3i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-594. - 1.02e3i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (2.95e4 - 5.12e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 1.43e4T + 1.80e9T^{2}$$
73 $$1 + 5.30e4T + 2.07e9T^{2}$$
79 $$1 + (-1.86e4 - 3.23e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-6.04e4 - 1.04e5i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + 9.78e4T + 5.58e9T^{2}$$
97 $$1 + (5.33e4 + 9.24e4i)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}