↑

Properties

Label 2-192-16.13-c3-0-6
Degree $2$
Conductor $192$
Sign $0.972 + 0.234i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 − 2.12i)3-s + (10.2 − 10.2i)5-s + 32.8i·7-s + 8.99i·9-s + (18.2 − 18.2i)11-s + (22.5 + 22.5i)13-s − 43.6·15-s + 50.1·17-s + (6.68 + 6.68i)19-s + (69.6 − 69.6i)21-s − 186. i·23-s − 86.9i·25-s + (19.0 − 19.0i)27-s + (118. + 118. i)29-s + 250.·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.920 − 0.920i)5-s + 1.77i·7-s + 0.333i·9-s + (0.499 − 0.499i)11-s + (0.481 + 0.481i)13-s − 0.751·15-s + 0.715·17-s + (0.0807 + 0.0807i)19-s + (0.723 − 0.723i)21-s − 1.69i·23-s − 0.695i·25-s + (0.136 − 0.136i)27-s + (0.757 + 0.757i)29-s + 1.44·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.86760 - 0.222282i\)
\(L(\frac12)\) \(\approx\) \(1.86760 - 0.222282i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-10.2 + 10.2i)T - 125iT^{2} \)
7 \( 1 - 32.8iT - 343T^{2} \)
11 \( 1 + (-18.2 + 18.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (-22.5 - 22.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 50.1T + 4.91e3T^{2} \)
19 \( 1 + (-6.68 - 6.68i)T + 6.85e3iT^{2} \)
23 \( 1 + 186. iT - 1.21e4T^{2} \)
29 \( 1 + (-118. - 118. i)T + 2.43e4iT^{2} \)
31 \( 1 - 250.T + 2.97e4T^{2} \)
37 \( 1 + (-198. + 198. i)T - 5.06e4iT^{2} \)
41 \( 1 - 186. iT - 6.89e4T^{2} \)
43 \( 1 + (10.9 - 10.9i)T - 7.95e4iT^{2} \)
47 \( 1 + 23.1T + 1.03e5T^{2} \)
53 \( 1 + (-134. + 134. i)T - 1.48e5iT^{2} \)
59 \( 1 + (220. - 220. i)T - 2.05e5iT^{2} \)
61 \( 1 + (453. + 453. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-184. - 184. i)T + 3.00e5iT^{2} \)
71 \( 1 + 18.8iT - 3.57e5T^{2} \)
73 \( 1 - 828. iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + (-173. - 173. i)T + 5.71e5iT^{2} \)
89 \( 1 + 335. iT - 7.04e5T^{2} \)
97 \( 1 + 687.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.25318701960811419585080361535, −11.32441601805881099253759135407, −9.894932166491974606330803058037, −8.892068785850589684225691500385, −8.367107948463225690239847888736, −6.40050629393269791190590940252, −5.82059503163988142842086285778, −4.79239307763165928644153129395, −2.59149774875863120990447689040, −1.22985904211299019388564127945, 1.17115417920123051463198753076, 3.23644530275392427162874076793, 4.42372222238006911923707267494, 5.92845821429848905568155069390, 6.83378606923692136195495399250, 7.81127465555261005198073397859, 9.674068626271713981187565542617, 10.14386689458572180662457316856, 10.87752542808031958816548644988, 11.91457306914759691722354951664

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