# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.849 + 0.526i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.29 − 3.78i)2-s + (−12.6 + 9.79i)4-s + (10.5 − 18.3i)5-s + (38.6 − 22.3i)7-s + (53.4 + 35.2i)8-s + (−83.0 − 16.3i)10-s + (58.6 − 33.8i)11-s + (14.5 − 25.2i)13-s + (−134. − 117. i)14-s + (64.2 − 247. i)16-s − 402.·17-s − 644. i·19-s + (45.5 + 335. i)20-s + (−204. − 178. i)22-s + (−335. − 193. i)23-s + ⋯
 L(s)  = 1 + (−0.323 − 0.946i)2-s + (−0.790 + 0.611i)4-s + (0.423 − 0.732i)5-s + (0.788 − 0.455i)7-s + (0.834 + 0.550i)8-s + (−0.830 − 0.163i)10-s + (0.485 − 0.280i)11-s + (0.0861 − 0.149i)13-s + (−0.685 − 0.599i)14-s + (0.251 − 0.967i)16-s − 1.39·17-s − 1.78i·19-s + (0.113 + 0.838i)20-s + (−0.421 − 0.368i)22-s + (−0.634 − 0.366i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.849 + 0.526i$ motivic weight = $$4$$ character : $\chi_{108} (91, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ -0.849 + 0.526i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.371845 - 1.30538i$$ $$L(\frac12)$$ $$\approx$$ $$0.371845 - 1.30538i$$ $$L(3)$$ 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 + (1.29 + 3.78i)T$$
3 $$1$$
good5 $$1 + (-10.5 + 18.3i)T + (-312.5 - 541. i)T^{2}$$
7 $$1 + (-38.6 + 22.3i)T + (1.20e3 - 2.07e3i)T^{2}$$
11 $$1 + (-58.6 + 33.8i)T + (7.32e3 - 1.26e4i)T^{2}$$
13 $$1 + (-14.5 + 25.2i)T + (-1.42e4 - 2.47e4i)T^{2}$$
17 $$1 + 402.T + 8.35e4T^{2}$$
19 $$1 + 644. iT - 1.30e5T^{2}$$
23 $$1 + (335. + 193. i)T + (1.39e5 + 2.42e5i)T^{2}$$
29 $$1 + (-362. - 627. i)T + (-3.53e5 + 6.12e5i)T^{2}$$
31 $$1 + (1.09e3 + 629. i)T + (4.61e5 + 7.99e5i)T^{2}$$
37 $$1 + 1.40e3T + 1.87e6T^{2}$$
41 $$1 + (-774. + 1.34e3i)T + (-1.41e6 - 2.44e6i)T^{2}$$
43 $$1 + (-1.62e3 + 935. i)T + (1.70e6 - 2.96e6i)T^{2}$$
47 $$1 + (-3.61e3 + 2.08e3i)T + (2.43e6 - 4.22e6i)T^{2}$$
53 $$1 + 906.T + 7.89e6T^{2}$$
59 $$1 + (-3.91e3 - 2.26e3i)T + (6.05e6 + 1.04e7i)T^{2}$$
61 $$1 + (1.31e3 + 2.27e3i)T + (-6.92e6 + 1.19e7i)T^{2}$$
67 $$1 + (58.7 + 33.8i)T + (1.00e7 + 1.74e7i)T^{2}$$
71 $$1 - 1.31e3iT - 2.54e7T^{2}$$
73 $$1 - 9.47e3T + 2.83e7T^{2}$$
79 $$1 + (3.78e3 - 2.18e3i)T + (1.94e7 - 3.37e7i)T^{2}$$
83 $$1 + (-659. + 381. i)T + (2.37e7 - 4.11e7i)T^{2}$$
89 $$1 + 8.08e3T + 6.27e7T^{2}$$
97 $$1 + (3.33e3 + 5.77e3i)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}