Properties

Degree $2$
Conductor $192$
Sign $0.726 + 0.687i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (1.69 − 1.69i)5-s + 5.74·7-s − 2.99i·9-s + (5.59 + 5.59i)11-s + (−13.5 − 13.5i)13-s − 4.16i·15-s + 19.7·17-s + (21.6 − 21.6i)19-s + (7.03 − 7.03i)21-s − 24.9·23-s + 19.2i·25-s + (−3.67 − 3.67i)27-s + (1.50 + 1.50i)29-s + 2.20i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.339 − 0.339i)5-s + 0.820·7-s − 0.333i·9-s + (0.508 + 0.508i)11-s + (−1.04 − 1.04i)13-s − 0.277i·15-s + 1.15·17-s + (1.14 − 1.14i)19-s + (0.334 − 0.334i)21-s − 1.08·23-s + 0.768i·25-s + (−0.136 − 0.136i)27-s + (0.0519 + 0.0519i)29-s + 0.0709i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.726 + 0.687i$
Motivic weight: \(2\)
Character: $\chi_{192} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.726 + 0.687i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.80379 - 0.718034i\)
\(L(\frac12)\) \(\approx\) \(1.80379 - 0.718034i\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
good5 \( 1 + (-1.69 + 1.69i)T - 25iT^{2} \)
7 \( 1 - 5.74T + 49T^{2} \)
11 \( 1 + (-5.59 - 5.59i)T + 121iT^{2} \)
13 \( 1 + (13.5 + 13.5i)T + 169iT^{2} \)
17 \( 1 - 19.7T + 289T^{2} \)
19 \( 1 + (-21.6 + 21.6i)T - 361iT^{2} \)
23 \( 1 + 24.9T + 529T^{2} \)
29 \( 1 + (-1.50 - 1.50i)T + 841iT^{2} \)
31 \( 1 - 2.20iT - 961T^{2} \)
37 \( 1 + (-27.6 + 27.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 51.3iT - 1.68e3T^{2} \)
43 \( 1 + (21.4 + 21.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 76.5iT - 2.20e3T^{2} \)
53 \( 1 + (56.5 - 56.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-48.0 - 48.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (51.5 + 51.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (63.4 - 63.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 43.4T + 5.04e3T^{2} \)
73 \( 1 - 73.9iT - 5.32e3T^{2} \)
79 \( 1 - 4.12iT - 6.24e3T^{2} \)
83 \( 1 + (38.4 - 38.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 52.9iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−12.27116236603601988649892398951, −11.40173716645377634829362583464, −9.986163166581230435605187024045, −9.274522895155907821618810733015, −7.922291582803701823095405703366, −7.36186560787187861588520572543, −5.73370262620505840261307968914, −4.69130062854786280716629153335, −2.91491663454029029810670624769, −1.32521575565122603045165404621, 1.86971880102786645922081865275, 3.49843320985271908252045059686, 4.81680572321576901431856270694, 6.05557824108513174711899916132, 7.47941956872004469345467952026, 8.361574790583027789890575129442, 9.649418644634103621436688347455, 10.19606153174793238023737012859, 11.60379136869704027620594655456, 12.12102298637039066591011473233

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