# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{4}$ Sign $-0.615 - 0.788i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.75 + 3.59i)2-s + (−9.84 − 12.6i)4-s + 46.6·5-s + 60.5i·7-s + (62.6 − 13.2i)8-s + (−81.9 + 167. i)10-s + 73.6i·11-s − 31.1·13-s + (−217. − 106. i)14-s + (−62.2 + 248. i)16-s + 53.8·17-s − 54.9i·19-s + (−459. − 589. i)20-s + (−264. − 129. i)22-s + 281. i·23-s + ⋯
 L(s)  = 1 + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + 1.86·5-s + 1.23i·7-s + (0.978 − 0.206i)8-s + (−0.819 + 1.67i)10-s + 0.608i·11-s − 0.184·13-s + (−1.11 − 0.542i)14-s + (−0.243 + 0.969i)16-s + 0.186·17-s − 0.152i·19-s + (−1.14 − 1.47i)20-s + (−0.546 − 0.266i)22-s + 0.532i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$324$$    =    $$2^{2} \cdot 3^{4}$$ $$\varepsilon$$ = $-0.615 - 0.788i$ motivic weight = $$4$$ character : $\chi_{324} (163, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 324,\ (\ :2),\ -0.615 - 0.788i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$2.007471817$$ $$L(\frac12)$$ $$\approx$$ $$2.007471817$$ $$L(3)$$ 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 + (1.75 - 3.59i)T$$
3 $$1$$
good5 $$1 - 46.6T + 625T^{2}$$
7 $$1 - 60.5iT - 2.40e3T^{2}$$
11 $$1 - 73.6iT - 1.46e4T^{2}$$
13 $$1 + 31.1T + 2.85e4T^{2}$$
17 $$1 - 53.8T + 8.35e4T^{2}$$
19 $$1 + 54.9iT - 1.30e5T^{2}$$
23 $$1 - 281. iT - 2.79e5T^{2}$$
29 $$1 + 447.T + 7.07e5T^{2}$$
31 $$1 - 277. iT - 9.23e5T^{2}$$
37 $$1 - 1.01e3T + 1.87e6T^{2}$$
41 $$1 - 1.89e3T + 2.82e6T^{2}$$
43 $$1 + 769. iT - 3.41e6T^{2}$$
47 $$1 - 2.74e3iT - 4.87e6T^{2}$$
53 $$1 + 4.64e3T + 7.89e6T^{2}$$
59 $$1 - 303. iT - 1.21e7T^{2}$$
61 $$1 - 956.T + 1.38e7T^{2}$$
67 $$1 - 6.94e3iT - 2.01e7T^{2}$$
71 $$1 + 5.97e3iT - 2.54e7T^{2}$$
73 $$1 + 4.33e3T + 2.83e7T^{2}$$
79 $$1 - 3.80e3iT - 3.89e7T^{2}$$
83 $$1 + 3.15e3iT - 4.74e7T^{2}$$
89 $$1 + 7.13e3T + 6.27e7T^{2}$$
97 $$1 + 1.96e3T + 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

−11.00989911531193486999927941714, −9.869243519664426269632954790279, −9.441059827792387776008855913960, −8.710975813829412226988793288475, −7.39733264953367177018717810519, −6.22098763272688644818112399987, −5.69853520096722251537511020577, −4.82191319360758231281827956072, −2.50375107498926837624772315299, −1.47147510413788594077423114865, 0.72393605848582869188956677459, 1.80151765294612462596472454507, 2.99609349841148681587356673717, 4.37427564050582932624456299720, 5.63008530192975342798278916201, 6.79531384239577347337532467724, 7.989866729164433461260193973613, 9.174363856202915637380133080926, 9.837501735718840780242456674227, 10.53040762082929532296942454185