# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.368 + 0.929i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (7.67 + 4.43i)5-s + (−30.9 − 53.5i)7-s + (−94.7 + 54.7i)11-s + (77.8 − 134. i)13-s − 395. i·17-s + 140.·19-s + (−802. − 463. i)23-s + (−273. − 473. i)25-s + (−323. + 186. i)29-s + (521. − 903. i)31-s − 548. i·35-s − 194.·37-s + (2.34e3 + 1.35e3i)41-s + (−167. − 290. i)43-s + (−2.46e3 + 1.42e3i)47-s + ⋯
 L(s)  = 1 + (0.307 + 0.177i)5-s + (−0.631 − 1.09i)7-s + (−0.783 + 0.452i)11-s + (0.460 − 0.798i)13-s − 1.37i·17-s + 0.388·19-s + (−1.51 − 0.876i)23-s + (−0.437 − 0.757i)25-s + (−0.385 + 0.222i)29-s + (0.543 − 0.940i)31-s − 0.447i·35-s − 0.142·37-s + (1.39 + 0.805i)41-s + (−0.0908 − 0.157i)43-s + (−1.11 + 0.645i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.368 + 0.929i$ motivic weight = $$4$$ character : $\chi_{108} (17, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ -0.368 + 0.929i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.627625 - 0.924119i$$ $$L(\frac12)$$ $$\approx$$ $$0.627625 - 0.924119i$$ $$L(3)$$ 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 + (-7.67 - 4.43i)T + (312.5 + 541. i)T^{2}$$
7 $$1 + (30.9 + 53.5i)T + (-1.20e3 + 2.07e3i)T^{2}$$
11 $$1 + (94.7 - 54.7i)T + (7.32e3 - 1.26e4i)T^{2}$$
13 $$1 + (-77.8 + 134. i)T + (-1.42e4 - 2.47e4i)T^{2}$$
17 $$1 + 395. iT - 8.35e4T^{2}$$
19 $$1 - 140.T + 1.30e5T^{2}$$
23 $$1 + (802. + 463. i)T + (1.39e5 + 2.42e5i)T^{2}$$
29 $$1 + (323. - 186. i)T + (3.53e5 - 6.12e5i)T^{2}$$
31 $$1 + (-521. + 903. i)T + (-4.61e5 - 7.99e5i)T^{2}$$
37 $$1 + 194.T + 1.87e6T^{2}$$
41 $$1 + (-2.34e3 - 1.35e3i)T + (1.41e6 + 2.44e6i)T^{2}$$
43 $$1 + (167. + 290. i)T + (-1.70e6 + 2.96e6i)T^{2}$$
47 $$1 + (2.46e3 - 1.42e3i)T + (2.43e6 - 4.22e6i)T^{2}$$
53 $$1 + 2.76e3iT - 7.89e6T^{2}$$
59 $$1 + (-4.35e3 - 2.51e3i)T + (6.05e6 + 1.04e7i)T^{2}$$
61 $$1 + (-3.52e3 - 6.10e3i)T + (-6.92e6 + 1.19e7i)T^{2}$$
67 $$1 + (3.43e3 - 5.95e3i)T + (-1.00e7 - 1.74e7i)T^{2}$$
71 $$1 - 821. iT - 2.54e7T^{2}$$
73 $$1 - 4.09e3T + 2.83e7T^{2}$$
79 $$1 + (3.78e3 + 6.55e3i)T + (-1.94e7 + 3.37e7i)T^{2}$$
83 $$1 + (6.77e3 - 3.91e3i)T + (2.37e7 - 4.11e7i)T^{2}$$
89 $$1 + 1.28e3iT - 6.27e7T^{2}$$
97 $$1 + (-1.89e3 - 3.27e3i)T + (-4.42e7 + 7.66e7i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}