# Properties

 Degree $2$ Conductor $192$ Sign $-0.333 - 0.942i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − 2.82i)3-s + 5.65i·5-s − 6·7-s + (−7.00 + 5.65i)9-s + 5.65i·11-s − 10·13-s + (16.0 − 5.65i)15-s + 22.6i·17-s − 2·19-s + (6 + 16.9i)21-s + 11.3i·23-s − 7.00·25-s + (23.0 + 14.1i)27-s + 16.9i·29-s − 22·31-s + ⋯
 L(s)  = 1 + (−0.333 − 0.942i)3-s + 1.13i·5-s − 0.857·7-s + (−0.777 + 0.628i)9-s + 0.514i·11-s − 0.769·13-s + (1.06 − 0.377i)15-s + 1.33i·17-s − 0.105·19-s + (0.285 + 0.808i)21-s + 0.491i·23-s − 0.280·25-s + (0.851 + 0.523i)27-s + 0.585i·29-s − 0.709·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $-0.333 - 0.942i$ Motivic weight: $$2$$ Character: $\chi_{192} (65, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :1),\ -0.333 - 0.942i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.351606 + 0.497246i$$ $$L(\frac12)$$ $$\approx$$ $$0.351606 + 0.497246i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (1 + 2.82i)T$$
good5 $$1 - 5.65iT - 25T^{2}$$
7 $$1 + 6T + 49T^{2}$$
11 $$1 - 5.65iT - 121T^{2}$$
13 $$1 + 10T + 169T^{2}$$
17 $$1 - 22.6iT - 289T^{2}$$
19 $$1 + 2T + 361T^{2}$$
23 $$1 - 11.3iT - 529T^{2}$$
29 $$1 - 16.9iT - 841T^{2}$$
31 $$1 + 22T + 961T^{2}$$
37 $$1 - 6T + 1.36e3T^{2}$$
41 $$1 + 33.9iT - 1.68e3T^{2}$$
43 $$1 + 82T + 1.84e3T^{2}$$
47 $$1 + 67.8iT - 2.20e3T^{2}$$
53 $$1 + 62.2iT - 2.80e3T^{2}$$
59 $$1 - 73.5iT - 3.48e3T^{2}$$
61 $$1 - 86T + 3.72e3T^{2}$$
67 $$1 + 2T + 4.48e3T^{2}$$
71 $$1 - 124. iT - 5.04e3T^{2}$$
73 $$1 - 82T + 5.32e3T^{2}$$
79 $$1 - 10T + 6.24e3T^{2}$$
83 $$1 + 73.5iT - 6.88e3T^{2}$$
89 $$1 - 33.9iT - 7.92e3T^{2}$$
97 $$1 + 94T + 9.40e3T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−12.65584047590198450828423606930, −11.72469184887904304875096522314, −10.66351050982243380319133115332, −9.877753904053585087025537825215, −8.380862154553730997341495473527, −7.09979244414894395021587982846, −6.68819988717016081537271772800, −5.46015142984845819772878767787, −3.46766842693641246302661146971, −2.11069408651266285238567965592, 0.34949447096554236819794990158, 3.06375294348636723508846077756, 4.51940446466197251972109600854, 5.35194265588145884900693417579, 6.59387133494967527884977702011, 8.203195787830959945570876474767, 9.340386222186197727321680144535, 9.727433502065386965565122863579, 11.04149073009512687228756808992, 12.01950614117946585443493170285