# Properties

 Label 2-192-4.3-c8-0-11 Degree $2$ Conductor $192$ Sign $-i$ Analytic cond. $78.2166$ Root an. cond. $8.84402$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 46.7i·3-s + 90·5-s − 921. i·7-s − 2.18e3·9-s + 5.17e3i·11-s + 1.63e4·13-s + 4.20e3i·15-s − 4.26e4·17-s − 1.14e5i·19-s + 4.30e4·21-s + 2.50e5i·23-s − 3.82e5·25-s − 1.02e5i·27-s + 1.27e6·29-s + 5.12e5i·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.144·5-s − 0.383i·7-s − 0.333·9-s + 0.353i·11-s + 0.572·13-s + 0.0831i·15-s − 0.510·17-s − 0.878i·19-s + 0.221·21-s + 0.895i·23-s − 0.979·25-s − 0.192i·27-s + 1.79·29-s + 0.555i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$192$$    =    $$2^{6} \cdot 3$$ Sign: $-i$ Analytic conductor: $$78.2166$$ Root analytic conductor: $$8.84402$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{192} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 192,\ (\ :4),\ -i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.811652174$$ $$L(\frac12)$$ $$\approx$$ $$1.811652174$$ $$L(5)$$ 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 - 46.7iT$$
good5 $$1 - 90T + 3.90e5T^{2}$$
7 $$1 + 921. iT - 5.76e6T^{2}$$
11 $$1 - 5.17e3iT - 2.14e8T^{2}$$
13 $$1 - 1.63e4T + 8.15e8T^{2}$$
17 $$1 + 4.26e4T + 6.97e9T^{2}$$
19 $$1 + 1.14e5iT - 1.69e10T^{2}$$
23 $$1 - 2.50e5iT - 7.83e10T^{2}$$
29 $$1 - 1.27e6T + 5.00e11T^{2}$$
31 $$1 - 5.12e5iT - 8.52e11T^{2}$$
37 $$1 - 2.26e6T + 3.51e12T^{2}$$
41 $$1 + 8.72e5T + 7.98e12T^{2}$$
43 $$1 + 1.66e6iT - 1.16e13T^{2}$$
47 $$1 + 7.97e5iT - 2.38e13T^{2}$$
53 $$1 + 1.06e6T + 6.22e13T^{2}$$
59 $$1 - 2.33e7iT - 1.46e14T^{2}$$
61 $$1 + 1.53e7T + 1.91e14T^{2}$$
67 $$1 - 9.18e6iT - 4.06e14T^{2}$$
71 $$1 + 2.28e7iT - 6.45e14T^{2}$$
73 $$1 - 1.89e7T + 8.06e14T^{2}$$
79 $$1 - 5.41e7iT - 1.51e15T^{2}$$
83 $$1 - 6.52e7iT - 2.25e15T^{2}$$
89 $$1 + 8.98e7T + 3.93e15T^{2}$$
97 $$1 + 7.57e7T + 7.83e15T^{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

−11.17663564458558892906729665674, −10.31493519699668690291686287187, −9.401717918600168033619578215770, −8.435221770366683672513051494272, −7.20810376766686490097042406009, −6.08503495102460841810916594249, −4.84522668961560352330140313436, −3.87562757559830239112091574006, −2.56096444042116705894423438857, −1.02725113268760986129137798978, 0.48186481734569619373559880614, 1.77223625393023371823959703720, 2.94189272429569635697947637394, 4.37989315803777759435320920620, 5.84501023397055437824002751095, 6.51101887454120145175148926990, 7.909163405288233887328323742015, 8.623849205015333757415804978052, 9.807115753948319195432182315175, 10.92331027106690402170520008737