# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.617 - 0.786i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.08 + 2.47i)2-s + (19.7 − 25.1i)4-s + 33.1i·5-s − 59.4i·7-s + (−38.4 + 176. i)8-s + (−81.9 − 168. i)10-s + 179.·11-s − 143.·13-s + (146. + 302. i)14-s + (−241. − 995. i)16-s − 93.8i·17-s − 1.61e3i·19-s + (834. + 655. i)20-s + (−911. + 442. i)22-s + 2.41e3·23-s + ⋯
 L(s)  = 1 + (−0.899 + 0.437i)2-s + (0.617 − 0.786i)4-s + 0.593i·5-s − 0.458i·7-s + (−0.212 + 0.977i)8-s + (−0.259 − 0.533i)10-s + 0.446·11-s − 0.235·13-s + (0.200 + 0.412i)14-s + (−0.236 − 0.971i)16-s − 0.0787i·17-s − 1.02i·19-s + (0.466 + 0.366i)20-s + (−0.401 + 0.195i)22-s + 0.951·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.617 - 0.786i$ motivic weight = $$5$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.617 - 0.786i)$$ $$L(3)$$ $$\approx$$ $$1.06936 + 0.519614i$$ $$L(\frac12)$$ $$\approx$$ $$1.06936 + 0.519614i$$ $$L(\frac{7}{2})$$ 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 + (5.08 - 2.47i)T$$
3 $$1$$
good5 $$1 - 33.1iT - 3.12e3T^{2}$$
7 $$1 + 59.4iT - 1.68e4T^{2}$$
11 $$1 - 179.T + 1.61e5T^{2}$$
13 $$1 + 143.T + 3.71e5T^{2}$$
17 $$1 + 93.8iT - 1.41e6T^{2}$$
19 $$1 + 1.61e3iT - 2.47e6T^{2}$$
23 $$1 - 2.41e3T + 6.43e6T^{2}$$
29 $$1 - 7.91e3iT - 2.05e7T^{2}$$
31 $$1 - 4.72e3iT - 2.86e7T^{2}$$
37 $$1 - 9.38e3T + 6.93e7T^{2}$$
41 $$1 - 1.18e4iT - 1.15e8T^{2}$$
43 $$1 - 1.95e4iT - 1.47e8T^{2}$$
47 $$1 - 7.37e3T + 2.29e8T^{2}$$
53 $$1 + 2.68e4iT - 4.18e8T^{2}$$
59 $$1 - 6.14e3T + 7.14e8T^{2}$$
61 $$1 - 5.52e4T + 8.44e8T^{2}$$
67 $$1 + 3.88e4iT - 1.35e9T^{2}$$
71 $$1 + 4.52e4T + 1.80e9T^{2}$$
73 $$1 - 2.10e4T + 2.07e9T^{2}$$
79 $$1 - 7.64e4iT - 3.07e9T^{2}$$
83 $$1 + 5.85e4T + 3.93e9T^{2}$$
89 $$1 + 1.85e4iT - 5.58e9T^{2}$$
97 $$1 - 1.37e5T + 8.58e9T^{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

−12.94946728524265174707178242723, −11.40076340670329054510947531406, −10.73475051443707785447393502991, −9.612949430915241164807867295702, −8.607037036551296909625732987518, −7.22177723289156436813602436500, −6.60896755354801764842954607258, −4.97096073065524571441525808144, −2.89337951459957598895076648727, −1.04263375538950741189301661293, 0.795006178458790688793342862122, 2.34409145973516800983621999572, 4.04819942227570358646266433904, 5.85862072058812399938109653733, 7.32815459830175702438061685179, 8.494476381345074459046841281371, 9.303715382468068232402810006504, 10.34990079445942988882358458371, 11.60339887791023764987824562745, 12.32352778086235626947088482840