Properties

Degree $2$
Conductor $192$
Sign $0.577 + 0.816i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − 2.44i)3-s + 2.82i·5-s + 10.3·7-s + (−2.99 + 8.48i)9-s − 14.6i·11-s + 6·13-s + (6.92 − 4.89i)15-s − 22.6i·17-s + 10.3·19-s + (−18 − 25.4i)21-s − 29.3i·23-s + 17·25-s + (25.9 − 7.34i)27-s + 31.1i·29-s − 31.1·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + 0.565i·5-s + 1.48·7-s + (−0.333 + 0.942i)9-s − 1.33i·11-s + 0.461·13-s + (0.461 − 0.326i)15-s − 1.33i·17-s + 0.546·19-s + (−0.857 − 1.21i)21-s − 1.27i·23-s + 0.680·25-s + (0.962 − 0.272i)27-s + 1.07i·29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.577 + 0.816i$
Motivic weight: \(2\)
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.27022 - 0.657516i\)
\(L(\frac12)\) \(\approx\) \(1.27022 - 0.657516i\)
\(L(2)\) 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 \( 1 + (1.73 + 2.44i)T \)
good5 \( 1 - 2.82iT - 25T^{2} \)
7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 - 6T + 169T^{2} \)
17 \( 1 + 22.6iT - 289T^{2} \)
19 \( 1 - 10.3T + 361T^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 - 31.1iT - 841T^{2} \)
31 \( 1 + 31.1T + 961T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 - 5.65iT - 1.68e3T^{2} \)
43 \( 1 - 10.3T + 1.84e3T^{2} \)
47 \( 1 - 58.7iT - 2.20e3T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + 14.6iT - 3.48e3T^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 + 114.T + 4.48e3T^{2} \)
71 \( 1 - 29.3iT - 5.04e3T^{2} \)
73 \( 1 + 30T + 5.32e3T^{2} \)
79 \( 1 + 31.1T + 6.24e3T^{2} \)
83 \( 1 - 73.4iT - 6.88e3T^{2} \)
89 \( 1 - 5.65iT - 7.92e3T^{2} \)
97 \( 1 - 90T + 9.40e3T^{2} \)
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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

−11.94139043721492925234238407346, −11.09140128726996742402026178701, −10.81295518720088555282981347994, −8.890019075588946499470974626715, −7.954762290277353143503316656915, −7.02679106083668334129796750170, −5.84481953356596687067487609140, −4.81896274423511660699511031808, −2.80087441125730779139323786691, −1.06871512151281810989255269874, 1.57108700097724275686614189191, 4.00362511082225894223277908158, 4.86792930215584232727671835704, 5.80458253610932666629022668597, 7.44175615020804701800707046341, 8.535448451387586839593442363822, 9.556183032904937557900245032665, 10.57408083106341655455679315324, 11.45296855248256347179285067826, 12.20244576266472459658654821740

Graph of the $Z$-function along the critical line