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Properties

Label 2-192-48.29-c2-0-9
Degree $2$
Conductor $192$
Sign $-0.784 + 0.619i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.50 + 2.59i)3-s + (−2.59 − 2.59i)5-s + 7.30i·7-s + (−4.47 − 7.81i)9-s + (−11.3 − 11.3i)11-s + (−0.746 − 0.746i)13-s + (10.6 − 2.83i)15-s + 6.67i·17-s + (−22.1 − 22.1i)19-s + (−18.9 − 10.9i)21-s − 21.4·23-s − 11.4i·25-s + (26.9 + 0.153i)27-s + (1.54 − 1.54i)29-s + 14.6·31-s + ⋯
L(s)  = 1  + (−0.501 + 0.865i)3-s + (−0.519 − 0.519i)5-s + 1.04i·7-s + (−0.496 − 0.867i)9-s + (−1.02 − 1.02i)11-s + (−0.0574 − 0.0574i)13-s + (0.710 − 0.188i)15-s + 0.392i·17-s + (−1.16 − 1.16i)19-s + (−0.902 − 0.523i)21-s − 0.932·23-s − 0.459i·25-s + (0.999 + 0.00567i)27-s + (0.0531 − 0.0531i)29-s + 0.471·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.784 + 0.619i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.784 + 0.619i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0199595 - 0.0574668i\)
\(L(\frac12)\) \(\approx\) \(0.0199595 - 0.0574668i\)
\(L(2)\) 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 \( 1 + (1.50 - 2.59i)T \)
good5 \( 1 + (2.59 + 2.59i)T + 25iT^{2} \)
7 \( 1 - 7.30iT - 49T^{2} \)
11 \( 1 + (11.3 + 11.3i)T + 121iT^{2} \)
13 \( 1 + (0.746 + 0.746i)T + 169iT^{2} \)
17 \( 1 - 6.67iT - 289T^{2} \)
19 \( 1 + (22.1 + 22.1i)T + 361iT^{2} \)
23 \( 1 + 21.4T + 529T^{2} \)
29 \( 1 + (-1.54 + 1.54i)T - 841iT^{2} \)
31 \( 1 - 14.6T + 961T^{2} \)
37 \( 1 + (50.1 - 50.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 15.0T + 1.68e3T^{2} \)
43 \( 1 + (26.3 - 26.3i)T - 1.84e3iT^{2} \)
47 \( 1 - 36.6iT - 2.20e3T^{2} \)
53 \( 1 + (-50.9 - 50.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-12.1 - 12.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (27.5 + 27.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (-4.84 - 4.84i)T + 4.48e3iT^{2} \)
71 \( 1 - 74.9T + 5.04e3T^{2} \)
73 \( 1 - 3.47iT - 5.32e3T^{2} \)
79 \( 1 + 103.T + 6.24e3T^{2} \)
83 \( 1 + (-31.7 + 31.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 78.2T + 7.92e3T^{2} \)
97 \( 1 + 61.5T + 9.40e3T^{2} \)
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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

−11.85916454632195013317061515104, −10.97250646435429641853452899480, −10.06425781638724712925009713370, −8.719245507695176174618257742197, −8.317121278449869047536652101115, −6.36163335654799945441023797087, −5.39199217719903318935537454063, −4.39064048104761317106465430002, −2.86210229878619399842785593979, −0.03529347098289165538934055700, 2.06684314280280469116456971391, 3.91859211682574712441168045412, 5.33301112477451320367640924511, 6.78422246042857713961275424468, 7.41684171539664872974183677038, 8.258947354807814049054489807798, 10.19441173026913699362040930353, 10.65133755010597097288506273522, 11.83097531186075364766070337225, 12.61766749049911583138723087632

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