Properties

Label 2-192-64.5-c1-0-14
Degree $2$
Conductor $192$
Sign $0.660 + 0.750i$
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  + (1.11 − 0.867i)2-s + (0.831 + 0.555i)3-s + (0.495 − 1.93i)4-s + (−0.668 − 0.133i)5-s + (1.41 − 0.100i)6-s + (0.440 − 1.06i)7-s + (−1.12 − 2.59i)8-s + (0.382 + 0.923i)9-s + (−0.862 + 0.431i)10-s + (1.16 + 1.73i)11-s + (1.48 − 1.33i)12-s + (1.13 − 0.226i)13-s + (−0.430 − 1.57i)14-s + (−0.482 − 0.482i)15-s + (−3.50 − 1.91i)16-s + (−4.47 + 4.47i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.613i)2-s + (0.480 + 0.320i)3-s + (0.247 − 0.968i)4-s + (−0.299 − 0.0594i)5-s + (0.575 − 0.0411i)6-s + (0.166 − 0.402i)7-s + (−0.398 − 0.917i)8-s + (0.127 + 0.307i)9-s + (−0.272 + 0.136i)10-s + (0.350 + 0.524i)11-s + (0.429 − 0.385i)12-s + (0.315 − 0.0628i)13-s + (−0.115 − 0.419i)14-s + (−0.124 − 0.124i)15-s + (−0.877 − 0.479i)16-s + (−1.08 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.660 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75123 - 0.791260i\)
\(L(\frac12)\) \(\approx\) \(1.75123 - 0.791260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.867i)T \)
3 \( 1 + (-0.831 - 0.555i)T \)
good5 \( 1 + (0.668 + 0.133i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-0.440 + 1.06i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.16 - 1.73i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (-1.13 + 0.226i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (4.47 - 4.47i)T - 17iT^{2} \)
19 \( 1 + (-0.731 - 3.67i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-1.46 + 0.606i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (3.43 - 5.13i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 6.02iT - 31T^{2} \)
37 \( 1 + (-0.203 + 1.02i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (6.26 - 2.59i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-7.89 + 5.27i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (-0.154 + 0.154i)T - 47iT^{2} \)
53 \( 1 + (4.50 + 6.74i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-5.69 - 1.13i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-9.02 - 6.02i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-1.77 - 1.18i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (-3.89 + 9.39i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.36 + 3.28i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.41 + 6.41i)T + 79iT^{2} \)
83 \( 1 + (-1.53 - 7.72i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (10.4 + 4.31i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−12.52627827985079904322416144826, −11.42411040157870610167537759299, −10.59106429294870853122408140209, −9.677064735922502221208054335461, −8.513758620005037026034741889224, −7.16265137607238018742804471665, −5.86290565660298517721204195975, −4.39420669930939381365839241658, −3.68381815581003315069006930050, −1.93269756001613951290123989118, 2.57729887035935901524415441700, 3.89428961316231455259008941638, 5.21940610658007416893355622352, 6.51010942252252630589834260591, 7.39473442248254918940881951669, 8.501542317699756415136445909454, 9.272230708161153969721119819855, 11.23706450114984942295239352434, 11.74157426733488188915730749978, 12.96919191269604254731107286424

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