↑

Properties

Label 2-192-192.59-c1-0-22
Degree $2$
Conductor $192$
Sign $0.245 + 0.969i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.364 + 1.36i)2-s + (−1.68 − 0.402i)3-s + (−1.73 + 0.995i)4-s + (0.701 − 3.52i)5-s + (−0.0631 − 2.44i)6-s + (−3.44 − 1.42i)7-s + (−1.99 − 2.00i)8-s + (2.67 + 1.35i)9-s + (5.07 − 0.325i)10-s + (−1.44 − 2.15i)11-s + (3.32 − 0.977i)12-s + (−2.02 + 0.402i)13-s + (0.695 − 5.22i)14-s + (−2.60 + 5.65i)15-s + (2.01 − 3.45i)16-s + (0.533 + 0.533i)17-s + ⋯
L(s)  = 1  + (0.257 + 0.966i)2-s + (−0.972 − 0.232i)3-s + (−0.867 + 0.497i)4-s + (0.313 − 1.57i)5-s + (−0.0257 − 0.999i)6-s + (−1.30 − 0.539i)7-s + (−0.704 − 0.710i)8-s + (0.891 + 0.452i)9-s + (1.60 − 0.102i)10-s + (−0.434 − 0.650i)11-s + (0.959 − 0.282i)12-s + (−0.561 + 0.111i)13-s + (0.185 − 1.39i)14-s + (−0.671 + 1.46i)15-s + (0.504 − 0.863i)16-s + (0.129 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449701 - 0.350046i\)
\(L(\frac12)\) \(\approx\) \(0.449701 - 0.350046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.364 - 1.36i)T \)
3 \( 1 + (1.68 + 0.402i)T \)
good5 \( 1 + (-0.701 + 3.52i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (3.44 + 1.42i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.44 + 2.15i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (2.02 - 0.402i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (-0.533 - 0.533i)T + 17iT^{2} \)
19 \( 1 + (-7.46 + 1.48i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (6.09 - 2.52i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (4.03 + 2.69i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 0.964T + 31T^{2} \)
37 \( 1 + (0.0846 - 0.425i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (1.77 + 4.29i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.02 - 1.53i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (-2.35 + 2.35i)T - 47iT^{2} \)
53 \( 1 + (-7.77 + 5.19i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (-7.01 - 1.39i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (0.760 + 0.508i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-6.24 + 9.34i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (0.666 - 1.60i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.36 - 10.5i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-5.18 + 5.18i)T - 79iT^{2} \)
83 \( 1 + (0.0562 + 0.282i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (0.727 - 1.75i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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

−12.54552432495801631591379417158, −11.81013356799687909427873870888, −9.960168315601244703491339980095, −9.427979763048275265169546160157, −8.038773743351249270796950634722, −7.02331193538919113190147089160, −5.78712174670582747752185643878, −5.23593119789476769959059781144, −3.89396551858834843189976228818, −0.51515023377744411374850848675, 2.53657220175302404519006783091, 3.64467457115995532052860526231, 5.36397591164980904957850263873, 6.23222156681185351837796952787, 7.32559203271647473740905815929, 9.627298476166483402034974965890, 9.921999832554049025683925370601, 10.72311467792822667643137876648, 11.82221999631058924288350847857, 12.41057840248763482781302232532

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