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Properties

Label 2-192-48.5-c2-0-3
Degree $2$
Conductor $192$
Sign $-0.276 - 0.961i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 + 0.737i)3-s + (1.57 − 1.57i)5-s + 3.64i·7-s + (7.91 − 4.29i)9-s + (−1.19 + 1.19i)11-s + (−14.6 + 14.6i)13-s + (−3.41 + 5.74i)15-s + 28.0i·17-s + (−12.5 + 12.5i)19-s + (−2.69 − 10.6i)21-s + 29.2·23-s + 20.0i·25-s + (−19.8 + 18.3i)27-s + (−19.3 − 19.3i)29-s − 11.6·31-s + ⋯
L(s)  = 1  + (−0.969 + 0.245i)3-s + (0.314 − 0.314i)5-s + 0.520i·7-s + (0.878 − 0.476i)9-s + (−0.108 + 0.108i)11-s + (−1.12 + 1.12i)13-s + (−0.227 + 0.382i)15-s + 1.65i·17-s + (−0.662 + 0.662i)19-s + (−0.128 − 0.504i)21-s + 1.27·23-s + 0.801i·25-s + (−0.734 + 0.678i)27-s + (−0.667 − 0.667i)29-s − 0.375·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.276 - 0.961i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.276 - 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.494142 + 0.656161i\)
\(L(\frac12)\) \(\approx\) \(0.494142 + 0.656161i\)
\(L(2)\) 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 \( 1 + (2.90 - 0.737i)T \)
good5 \( 1 + (-1.57 + 1.57i)T - 25iT^{2} \)
7 \( 1 - 3.64iT - 49T^{2} \)
11 \( 1 + (1.19 - 1.19i)T - 121iT^{2} \)
13 \( 1 + (14.6 - 14.6i)T - 169iT^{2} \)
17 \( 1 - 28.0iT - 289T^{2} \)
19 \( 1 + (12.5 - 12.5i)T - 361iT^{2} \)
23 \( 1 - 29.2T + 529T^{2} \)
29 \( 1 + (19.3 + 19.3i)T + 841iT^{2} \)
31 \( 1 + 11.6T + 961T^{2} \)
37 \( 1 + (-0.771 - 0.771i)T + 1.36e3iT^{2} \)
41 \( 1 + 25.6T + 1.68e3T^{2} \)
43 \( 1 + (-40.5 - 40.5i)T + 1.84e3iT^{2} \)
47 \( 1 + 50.2iT - 2.20e3T^{2} \)
53 \( 1 + (-46.2 + 46.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (22.7 - 22.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (-12.7 + 12.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 122.T + 5.04e3T^{2} \)
73 \( 1 + 15.0iT - 5.32e3T^{2} \)
79 \( 1 - 51.3T + 6.24e3T^{2} \)
83 \( 1 + (-37.8 - 37.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 5.45T + 7.92e3T^{2} \)
97 \( 1 + 81.1T + 9.40e3T^{2} \)
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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

−12.50590682783813221240027597791, −11.67288589771642655442882795830, −10.67011804565854627578870792035, −9.709176456884568631446005601323, −8.803127105129405351159121273029, −7.27570281751374630005469028393, −6.16151369715554013142272405669, −5.21362026340395800426310175921, −4.06399266266121843652178026243, −1.83654079841516906502466647365, 0.53285264370671593875849076703, 2.69291988703037363756764567971, 4.67586827216423457361549689884, 5.56327729405500351497588658166, 6.95253383108946801632185667547, 7.49089693022083141753353579448, 9.212062118998842364775689651307, 10.34801297383506968498162904885, 10.91285748750018535120623087511, 12.03040874871685954268510630018

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