# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.900 + 0.434i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.68 + 1.56i)2-s + (11.1 − 11.5i)4-s + (−1.01 − 1.75i)5-s + (20.0 + 11.5i)7-s + (−22.8 + 59.7i)8-s + (6.48 + 4.88i)10-s + (−4.32 − 2.49i)11-s + (−137. − 238. i)13-s + (−91.8 − 11.2i)14-s + (−9.35 − 255. i)16-s + 266.·17-s + 367. i·19-s + (−31.5 − 7.83i)20-s + (19.8 + 2.42i)22-s + (544. − 314. i)23-s + ⋯
 L(s)  = 1 + (−0.920 + 0.391i)2-s + (0.694 − 0.719i)4-s + (−0.0406 − 0.0703i)5-s + (0.408 + 0.236i)7-s + (−0.357 + 0.934i)8-s + (0.0648 + 0.0488i)10-s + (−0.0357 − 0.0206i)11-s + (−0.815 − 1.41i)13-s + (−0.468 − 0.0573i)14-s + (−0.0365 − 0.999i)16-s + 0.920·17-s + 1.01i·19-s + (−0.0788 − 0.0195i)20-s + (0.0410 + 0.00501i)22-s + (1.02 − 0.594i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.900 + 0.434i$ motivic weight = $$4$$ character : $\chi_{108} (19, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.900 + 0.434i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.07275 - 0.245262i$$ $$L(\frac12)$$ $$\approx$$ $$1.07275 - 0.245262i$$ $$L(3)$$ 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 + (3.68 - 1.56i)T$$
3 $$1$$
good5 $$1 + (1.01 + 1.75i)T + (-312.5 + 541. i)T^{2}$$
7 $$1 + (-20.0 - 11.5i)T + (1.20e3 + 2.07e3i)T^{2}$$
11 $$1 + (4.32 + 2.49i)T + (7.32e3 + 1.26e4i)T^{2}$$
13 $$1 + (137. + 238. i)T + (-1.42e4 + 2.47e4i)T^{2}$$
17 $$1 - 266.T + 8.35e4T^{2}$$
19 $$1 - 367. iT - 1.30e5T^{2}$$
23 $$1 + (-544. + 314. i)T + (1.39e5 - 2.42e5i)T^{2}$$
29 $$1 + (-319. + 553. i)T + (-3.53e5 - 6.12e5i)T^{2}$$
31 $$1 + (-1.19e3 + 687. i)T + (4.61e5 - 7.99e5i)T^{2}$$
37 $$1 - 1.46e3T + 1.87e6T^{2}$$
41 $$1 + (593. + 1.02e3i)T + (-1.41e6 + 2.44e6i)T^{2}$$
43 $$1 + (-1.43e3 - 825. i)T + (1.70e6 + 2.96e6i)T^{2}$$
47 $$1 + (307. + 177. i)T + (2.43e6 + 4.22e6i)T^{2}$$
53 $$1 + 5.29e3T + 7.89e6T^{2}$$
59 $$1 + (-5.22e3 + 3.01e3i)T + (6.05e6 - 1.04e7i)T^{2}$$
61 $$1 + (833. - 1.44e3i)T + (-6.92e6 - 1.19e7i)T^{2}$$
67 $$1 + (1.90e3 - 1.10e3i)T + (1.00e7 - 1.74e7i)T^{2}$$
71 $$1 - 524. iT - 2.54e7T^{2}$$
73 $$1 + 1.49e3T + 2.83e7T^{2}$$
79 $$1 + (-4.44e3 - 2.56e3i)T + (1.94e7 + 3.37e7i)T^{2}$$
83 $$1 + (6.91e3 + 3.99e3i)T + (2.37e7 + 4.11e7i)T^{2}$$
89 $$1 - 8.86e3T + 6.27e7T^{2}$$
97 $$1 + (3.40e3 - 5.90e3i)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}