# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.612 + 0.790i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.93 − 2.06i)2-s + (−0.538 − 7.98i)4-s + (4.71 − 2.72i)5-s + (−20.9 − 12.0i)7-s + (−17.5 − 14.3i)8-s + (3.48 − 14.9i)10-s + (25.3 − 43.9i)11-s + (25.0 + 43.4i)13-s + (−65.4 + 19.9i)14-s + (−63.4 + 8.60i)16-s − 51.7i·17-s − 27.9i·19-s + (−24.2 − 36.1i)20-s + (−41.8 − 137. i)22-s + (−3.93 − 6.81i)23-s + ⋯
 L(s)  = 1 + (0.682 − 0.730i)2-s + (−0.0673 − 0.997i)4-s + (0.421 − 0.243i)5-s + (−1.13 − 0.652i)7-s + (−0.774 − 0.632i)8-s + (0.110 − 0.474i)10-s + (0.696 − 1.20i)11-s + (0.535 + 0.927i)13-s + (−1.24 + 0.380i)14-s + (−0.990 + 0.134i)16-s − 0.737i·17-s − 0.337i·19-s + (−0.271 − 0.404i)20-s + (−0.405 − 1.33i)22-s + (−0.0356 − 0.0617i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.612 + 0.790i$ motivic weight = $$3$$ character : $\chi_{108} (35, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -0.612 + 0.790i)$$ $$L(2)$$ $$\approx$$ $$0.890455 - 1.81547i$$ $$L(\frac12)$$ $$\approx$$ $$0.890455 - 1.81547i$$ $$L(\frac{5}{2})$$ 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 + (-1.93 + 2.06i)T$$
3 $$1$$
good5 $$1 + (-4.71 + 2.72i)T + (62.5 - 108. i)T^{2}$$
7 $$1 + (20.9 + 12.0i)T + (171.5 + 297. i)T^{2}$$
11 $$1 + (-25.3 + 43.9i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-25.0 - 43.4i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 + 51.7iT - 4.91e3T^{2}$$
19 $$1 + 27.9iT - 6.85e3T^{2}$$
23 $$1 + (3.93 + 6.81i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (-212. - 122. i)T + (1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-51.4 + 29.6i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 - 295.T + 5.06e4T^{2}$$
41 $$1 + (-146. + 84.7i)T + (3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-284. - 164. i)T + (3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (47.9 - 83.0i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + 300. iT - 1.48e5T^{2}$$
59 $$1 + (-113. - 196. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-173. + 300. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (904. - 522. i)T + (1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 243.T + 3.57e5T^{2}$$
73 $$1 + 1.09e3T + 3.89e5T^{2}$$
79 $$1 + (-530. - 306. i)T + (2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (-283. + 490. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 212. iT - 7.04e5T^{2}$$
97 $$1 + (-234. + 405. i)T + (-4.56e5 - 7.90e5i)T^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−13.06775843872980427649189783238, −11.79623893019305300802965495038, −10.92577190041303117391683844547, −9.709138527827778754303267391234, −8.970461410337169159811043019100, −6.77270057972626462021200361771, −5.91408430564476792560438587659, −4.25679272875190858446038039496, −3.04414512828299490666287421684, −0.958069253197356653259170408975, 2.70361994164753761677799388489, 4.19048908106014153658427575594, 5.89966660855544089632787203422, 6.49052714580195235265132838775, 7.919515787129410496564765560683, 9.213403570269077616368874638893, 10.26305198958773583163119823537, 12.02156967750254687163352506065, 12.66006690032844905452119836216, 13.58179366904546836448200377534