Properties

Label 2-192-192.131-c1-0-3
Degree $2$
Conductor $192$
Sign $0.447 - 0.894i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.194 − 1.40i)2-s + (−1.72 + 0.119i)3-s + (−1.92 − 0.544i)4-s + (−1.28 + 1.92i)5-s + (−0.167 + 2.44i)6-s + (1.24 + 2.99i)7-s + (−1.13 + 2.58i)8-s + (2.97 − 0.414i)9-s + (2.44 + 2.17i)10-s + (−4.09 + 0.814i)11-s + (3.39 + 0.710i)12-s + (2.40 − 1.60i)13-s + (4.44 − 1.15i)14-s + (1.99 − 3.48i)15-s + (3.40 + 2.09i)16-s + (−2.79 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.137 − 0.990i)2-s + (−0.997 + 0.0692i)3-s + (−0.962 − 0.272i)4-s + (−0.575 + 0.861i)5-s + (−0.0685 + 0.997i)6-s + (0.469 + 1.13i)7-s + (−0.401 + 0.915i)8-s + (0.990 − 0.138i)9-s + (0.774 + 0.688i)10-s + (−1.23 + 0.245i)11-s + (0.978 + 0.205i)12-s + (0.666 − 0.445i)13-s + (1.18 − 0.309i)14-s + (0.514 − 0.899i)15-s + (0.851 + 0.523i)16-s + (−0.678 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446015 + 0.275706i\)
\(L(\frac12)\) \(\approx\) \(0.446015 + 0.275706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.194 + 1.40i)T \)
3 \( 1 + (1.72 - 0.119i)T \)
good5 \( 1 + (1.28 - 1.92i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-1.24 - 2.99i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.09 - 0.814i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.40 + 1.60i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (2.79 - 2.79i)T - 17iT^{2} \)
19 \( 1 + (4.13 - 2.76i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (0.248 - 0.600i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.476 - 2.39i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 4.97T + 31T^{2} \)
37 \( 1 + (-1.60 + 2.39i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-6.25 - 2.58i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-9.47 + 1.88i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (5.26 + 5.26i)T + 47iT^{2} \)
53 \( 1 + (2.26 + 11.4i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-8.56 - 5.72i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (2.21 - 11.1i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (9.19 + 1.82i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-9.64 + 3.99i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (10.4 + 4.34i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.55 - 4.55i)T + 79iT^{2} \)
83 \( 1 + (-6.97 - 10.4i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-0.570 + 0.236i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−12.58073640289898796097550966101, −11.54262033916223953491844426175, −10.86217189574685127067896668836, −10.35374017054896996254329232421, −8.857514462507115602869201174468, −7.73612927660096036823790556511, −6.08338826978124899616185663529, −5.18881152668368564584194169806, −3.83032172014651455588468893739, −2.22472432277995637927867003538, 0.50521806534453270712294693498, 4.27350424828027960135176531352, 4.75115487016381161177718282487, 6.04897999878831610540763356666, 7.23812716065512499394275721742, 7.991899985824346955701860431847, 9.140403189513502224323064620508, 10.58381940035136198903799871932, 11.32730338198519059856714775918, 12.71670954696635108920484337955

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