# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.267 - 0.963i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.270 − 1.98i)2-s + (−3.85 + 1.07i)4-s − 7.40·5-s + 9.47i·7-s + (3.16 + 7.34i)8-s + (2.00 + 14.6i)10-s − 6.77i·11-s − 14.4·13-s + (18.7 − 2.55i)14-s + (13.7 − 8.25i)16-s − 17.8·17-s − 5.19i·19-s + (28.5 − 7.92i)20-s + (−13.4 + 1.82i)22-s + 24.9i·23-s + ⋯
 L(s)  = 1 + (−0.135 − 0.990i)2-s + (−0.963 + 0.267i)4-s − 1.48·5-s + 1.35i·7-s + (0.395 + 0.918i)8-s + (0.200 + 1.46i)10-s − 0.615i·11-s − 1.10·13-s + (1.34 − 0.182i)14-s + (0.856 − 0.515i)16-s − 1.05·17-s − 0.273i·19-s + (1.42 − 0.396i)20-s + (−0.609 + 0.0831i)22-s + 1.08i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.267 - 0.963i$ motivic weight = $$2$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :1),\ -0.267 - 0.963i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.120338 + 0.158317i$$ $$L(\frac12)$$ $$\approx$$ $$0.120338 + 0.158317i$$ $$L(2)$$ 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 + (0.270 + 1.98i)T$$
3 $$1$$
good5 $$1 + 7.40T + 25T^{2}$$
7 $$1 - 9.47iT - 49T^{2}$$
11 $$1 + 6.77iT - 121T^{2}$$
13 $$1 + 14.4T + 169T^{2}$$
17 $$1 + 17.8T + 289T^{2}$$
19 $$1 + 5.19iT - 361T^{2}$$
23 $$1 - 24.9iT - 529T^{2}$$
29 $$1 + 29.6T + 841T^{2}$$
31 $$1 + 17.1iT - 961T^{2}$$
37 $$1 - 6.41T + 1.36e3T^{2}$$
41 $$1 + 8.64T + 1.68e3T^{2}$$
43 $$1 + 50.1iT - 1.84e3T^{2}$$
47 $$1 - 52.0iT - 2.20e3T^{2}$$
53 $$1 - 82.6T + 2.80e3T^{2}$$
59 $$1 - 83.7iT - 3.48e3T^{2}$$
61 $$1 + 8.41T + 3.72e3T^{2}$$
67 $$1 - 29.0iT - 4.48e3T^{2}$$
71 $$1 + 113. iT - 5.04e3T^{2}$$
73 $$1 - 2.16T + 5.32e3T^{2}$$
79 $$1 - 149. iT - 6.24e3T^{2}$$
83 $$1 - 13.5iT - 6.88e3T^{2}$$
89 $$1 + 17.8T + 7.92e3T^{2}$$
97 $$1 + 138.T + 9.40e3T^{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

−13.44877313212108002196802733450, −12.31079871509416210168286102598, −11.72476357320115904484384615050, −10.99187901405261310237159377195, −9.395941384890138234661744269566, −8.574837322348198668940840750848, −7.48694990637604822106887920300, −5.37641742250104891724606986686, −3.99940642699208796017580529048, −2.57708406636371619572450183894, 0.15036262227245284629706915437, 3.96386930486340945850207757335, 4.73643627113021428583584722420, 6.87707821198327328985652088988, 7.41965267939233516901188910055, 8.417495319166231447217197702458, 9.857146379377837818732311365967, 10.92736028792514873252453918928, 12.28512698911986900601552598046, 13.29180144644114077499245994239