# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{4}$ Sign $-0.257 + 0.966i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (2.43 + 3.17i)2-s + (−4.11 + 15.4i)4-s − 22.1·5-s + 95.5i·7-s + (−59.0 + 24.6i)8-s + (−54.0 − 70.2i)10-s − 21.8i·11-s + 126.·13-s + (−303. + 233. i)14-s + (−222. − 127. i)16-s − 283.·17-s + 323. i·19-s + (91.0 − 342. i)20-s + (69.3 − 53.3i)22-s − 229. i·23-s + ⋯
 L(s)  = 1 + (0.609 + 0.792i)2-s + (−0.257 + 0.966i)4-s − 0.885·5-s + 1.95i·7-s + (−0.922 + 0.385i)8-s + (−0.540 − 0.702i)10-s − 0.180i·11-s + 0.746·13-s + (−1.54 + 1.18i)14-s + (−0.867 − 0.496i)16-s − 0.982·17-s + 0.896i·19-s + (0.227 − 0.856i)20-s + (0.143 − 0.110i)22-s − 0.433i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$324$$    =    $$2^{2} \cdot 3^{4}$$ $$\varepsilon$$ = $-0.257 + 0.966i$ motivic weight = $$4$$ character : $\chi_{324} (163, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 324,\ (\ :2),\ -0.257 + 0.966i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.8535999729$$ $$L(\frac12)$$ $$\approx$$ $$0.8535999729$$ $$L(3)$$ 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 + (-2.43 - 3.17i)T$$
3 $$1$$
good5 $$1 + 22.1T + 625T^{2}$$
7 $$1 - 95.5iT - 2.40e3T^{2}$$
11 $$1 + 21.8iT - 1.46e4T^{2}$$
13 $$1 - 126.T + 2.85e4T^{2}$$
17 $$1 + 283.T + 8.35e4T^{2}$$
19 $$1 - 323. iT - 1.30e5T^{2}$$
23 $$1 + 229. iT - 2.79e5T^{2}$$
29 $$1 - 1.20e3T + 7.07e5T^{2}$$
31 $$1 + 829. iT - 9.23e5T^{2}$$
37 $$1 + 318.T + 1.87e6T^{2}$$
41 $$1 - 328.T + 2.82e6T^{2}$$
43 $$1 + 207. iT - 3.41e6T^{2}$$
47 $$1 + 1.22e3iT - 4.87e6T^{2}$$
53 $$1 + 2.83e3T + 7.89e6T^{2}$$
59 $$1 + 1.47e3iT - 1.21e7T^{2}$$
61 $$1 + 1.87e3T + 1.38e7T^{2}$$
67 $$1 + 247. iT - 2.01e7T^{2}$$
71 $$1 + 4.30e3iT - 2.54e7T^{2}$$
73 $$1 - 3.01e3T + 2.83e7T^{2}$$
79 $$1 - 7.19e3iT - 3.89e7T^{2}$$
83 $$1 - 3.32e3iT - 4.74e7T^{2}$$
89 $$1 + 1.54e3T + 6.27e7T^{2}$$
97 $$1 - 5.83e3T + 8.85e7T^{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

−11.94947785676057817656285440586, −11.09485899283874428394500294012, −9.364949721352874688302451210166, −8.462531818742587492227316353292, −8.048577898604589707695581319941, −6.54257423866667934816357578440, −5.86108474630017327097855693998, −4.75952965552699647632178589835, −3.59508750022029400396155459478, −2.39727460151719727915908501105, 0.21912276138223230490591733842, 1.29918758836951178928226739615, 3.16012556737665008125377375509, 4.12934482443382692509611963958, 4.70384486678127707249112584827, 6.45904756069060735232881119092, 7.26200510672943273786238779052, 8.462450430282631583743973080991, 9.700387663707605557917602233323, 10.75327706077385135598896794860