↑

Properties

Label 2-192-24.11-c7-0-37
Degree $2$
Conductor $192$
Sign $0.867 - 0.496i$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (45.1 + 12.2i)3-s + 464.·5-s − 639. i·7-s + (1.88e3 + 1.10e3i)9-s − 1.35e3i·11-s + 1.04e4i·13-s + (2.09e4 + 5.69e3i)15-s + 2.64e4i·17-s − 1.84e4·19-s + (7.83e3 − 2.88e4i)21-s + 8.61e4·23-s + 1.37e5·25-s + (7.15e4 + 7.31e4i)27-s − 5.92e4·29-s + 1.22e5i·31-s + ⋯
L(s)  = 1  + (0.964 + 0.262i)3-s + 1.66·5-s − 0.704i·7-s + (0.862 + 0.506i)9-s − 0.306i·11-s + 1.31i·13-s + (1.60 + 0.435i)15-s + 1.30i·17-s − 0.617·19-s + (0.184 − 0.679i)21-s + 1.47·23-s + 1.75·25-s + (0.699 + 0.714i)27-s − 0.450·29-s + 0.740i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.867 - 0.496i$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ 0.867 - 0.496i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.475838590\)
\(L(\frac12)\) \(\approx\) \(4.475838590\)
\(L(\frac{9}{2})\) 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 \( 1 + (-45.1 - 12.2i)T \)
good5 \( 1 - 464.T + 7.81e4T^{2} \)
7 \( 1 + 639. iT - 8.23e5T^{2} \)
11 \( 1 + 1.35e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.04e4iT - 6.27e7T^{2} \)
17 \( 1 - 2.64e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.84e4T + 8.93e8T^{2} \)
23 \( 1 - 8.61e4T + 3.40e9T^{2} \)
29 \( 1 + 5.92e4T + 1.72e10T^{2} \)
31 \( 1 - 1.22e5iT - 2.75e10T^{2} \)
37 \( 1 - 8.41e4iT - 9.49e10T^{2} \)
41 \( 1 + 8.14e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.65e5T + 2.71e11T^{2} \)
47 \( 1 + 4.89e5T + 5.06e11T^{2} \)
53 \( 1 + 1.46e6T + 1.17e12T^{2} \)
59 \( 1 + 2.35e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.80e6iT - 3.14e12T^{2} \)
67 \( 1 - 3.61e6T + 6.06e12T^{2} \)
71 \( 1 - 1.92e6T + 9.09e12T^{2} \)
73 \( 1 + 3.70e6T + 1.10e13T^{2} \)
79 \( 1 + 4.55e6iT - 1.92e13T^{2} \)
83 \( 1 + 5.74e6iT - 2.71e13T^{2} \)
89 \( 1 + 9.67e6iT - 4.42e13T^{2} \)
97 \( 1 + 3.10e6T + 8.07e13T^{2} \)
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

−10.89780948007725420967128337417, −10.25722627614010068275205636625, −9.250998131251714682198695702062, −8.679284335769777919369584464945, −7.16408806419640975552453561051, −6.23910971572025524559964940253, −4.84009335743626338708178349655, −3.62573903780268919173061470329, −2.20102309780168073853995535776, −1.40881850938316095513896204216, 1.06214615745796213065543245647, 2.32870752450176553999286519339, 2.95158943413621750313342802845, 4.92888486111044445263989559635, 5.93964314457493061776985886095, 7.06379029095196201687258936606, 8.296143630646386749850744005966, 9.380300430258323352664975590420, 9.736843908979974108640176554881, 11.00306595130473010477443065601

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