Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.997 + 0.0640i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (5.65 + 0.181i)2-s + (31.9 + 2.04i)4-s − 69.1i·5-s + 238. i·7-s + (180. + 17.3i)8-s + (12.5 − 391. i)10-s + 350.·11-s + 669.·13-s + (−43.2 + 1.35e3i)14-s + (1.01e3 + 130. i)16-s − 1.39e3i·17-s − 1.78e3i·19-s + (141. − 2.20e3i)20-s + (1.98e3 + 63.5i)22-s + 1.32e3·23-s + ⋯
 L(s)  = 1 + (0.999 + 0.0320i)2-s + (0.997 + 0.0640i)4-s − 1.23i·5-s + 1.84i·7-s + (0.995 + 0.0959i)8-s + (0.0396 − 1.23i)10-s + 0.874·11-s + 1.09·13-s + (−0.0589 + 1.84i)14-s + (0.991 + 0.127i)16-s − 1.17i·17-s − 1.13i·19-s + (0.0792 − 1.23i)20-s + (0.874 + 0.0280i)22-s + 0.521·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0640i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.997 + 0.0640i$ motivic weight = $$5$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ 0.997 + 0.0640i)$$ $$L(3)$$ $$\approx$$ $$3.88773 - 0.124540i$$ $$L(\frac12)$$ $$\approx$$ $$3.88773 - 0.124540i$$ $$L(\frac{7}{2})$$ 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 + (-5.65 - 0.181i)T$$
3 $$1$$
good5 $$1 + 69.1iT - 3.12e3T^{2}$$
7 $$1 - 238. iT - 1.68e4T^{2}$$
11 $$1 - 350.T + 1.61e5T^{2}$$
13 $$1 - 669.T + 3.71e5T^{2}$$
17 $$1 + 1.39e3iT - 1.41e6T^{2}$$
19 $$1 + 1.78e3iT - 2.47e6T^{2}$$
23 $$1 - 1.32e3T + 6.43e6T^{2}$$
29 $$1 - 6.89e3iT - 2.05e7T^{2}$$
31 $$1 - 2.33e3iT - 2.86e7T^{2}$$
37 $$1 + 1.29e4T + 6.93e7T^{2}$$
41 $$1 + 8.71e3iT - 1.15e8T^{2}$$
43 $$1 - 1.06e4iT - 1.47e8T^{2}$$
47 $$1 + 3.98e3T + 2.29e8T^{2}$$
53 $$1 - 4.05e3iT - 4.18e8T^{2}$$
59 $$1 + 2.51e4T + 7.14e8T^{2}$$
61 $$1 - 8.96e3T + 8.44e8T^{2}$$
67 $$1 - 1.24e4iT - 1.35e9T^{2}$$
71 $$1 - 6.50e3T + 1.80e9T^{2}$$
73 $$1 + 4.58e4T + 2.07e9T^{2}$$
79 $$1 + 3.74e4iT - 3.07e9T^{2}$$
83 $$1 + 4.86e4T + 3.93e9T^{2}$$
89 $$1 - 7.93e3iT - 5.58e9T^{2}$$
97 $$1 - 5.65e4T + 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

−12.64146287524481418258946923235, −12.01839135867323527563015369908, −11.16094294288877333883056731228, −9.119820615016479380255199557704, −8.671694100271282374246257132383, −6.79508224660000946298887067032, −5.53513187224664511499429868111, −4.80338758885235877119432943017, −3.06559961341032869074544483724, −1.46142461243087437656755537886, 1.46311192804971778677582882493, 3.51478916354474711398975880079, 4.05675843941955669945100178119, 6.15760853036413060387612516541, 6.84828809707850081049895324848, 7.927718512966526734643578688256, 10.20200506554184684945751553126, 10.72188787507600999306907467067, 11.61871024806270264443372947356, 13.07293986350509916374210508499