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Properties

Label 2-192-48.35-c3-0-2
Degree $2$
Conductor $192$
Sign $0.606 - 0.795i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.07 − 1.13i)3-s + (−11.2 − 11.2i)5-s − 30.2·7-s + (24.4 + 11.5i)9-s + (14.9 − 14.9i)11-s + (10.4 + 10.4i)13-s + (44.3 + 69.9i)15-s + 69.5i·17-s + (36.4 − 36.4i)19-s + (153. + 34.2i)21-s − 1.28i·23-s + 128. i·25-s + (−110. − 86.1i)27-s + (119. − 119. i)29-s + 172. i·31-s + ⋯
L(s)  = 1  + (−0.975 − 0.218i)3-s + (−1.00 − 1.00i)5-s − 1.63·7-s + (0.904 + 0.426i)9-s + (0.410 − 0.410i)11-s + (0.223 + 0.223i)13-s + (0.763 + 1.20i)15-s + 0.992i·17-s + (0.439 − 0.439i)19-s + (1.59 + 0.356i)21-s − 0.0116i·23-s + 1.03i·25-s + (−0.789 − 0.613i)27-s + (0.764 − 0.764i)29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.606 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.399778 + 0.198005i\)
\(L(\frac12)\) \(\approx\) \(0.399778 + 0.198005i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.07 + 1.13i)T \)
good5 \( 1 + (11.2 + 11.2i)T + 125iT^{2} \)
7 \( 1 + 30.2T + 343T^{2} \)
11 \( 1 + (-14.9 + 14.9i)T - 1.33e3iT^{2} \)
13 \( 1 + (-10.4 - 10.4i)T + 2.19e3iT^{2} \)
17 \( 1 - 69.5iT - 4.91e3T^{2} \)
19 \( 1 + (-36.4 + 36.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 1.28iT - 1.21e4T^{2} \)
29 \( 1 + (-119. + 119. i)T - 2.43e4iT^{2} \)
31 \( 1 - 172. iT - 2.97e4T^{2} \)
37 \( 1 + (235. - 235. i)T - 5.06e4iT^{2} \)
41 \( 1 + 19.9T + 6.89e4T^{2} \)
43 \( 1 + (-294. - 294. i)T + 7.95e4iT^{2} \)
47 \( 1 - 69.5T + 1.03e5T^{2} \)
53 \( 1 + (122. + 122. i)T + 1.48e5iT^{2} \)
59 \( 1 + (63.2 - 63.2i)T - 2.05e5iT^{2} \)
61 \( 1 + (352. + 352. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-228. + 228. i)T - 3.00e5iT^{2} \)
71 \( 1 + 524. iT - 3.57e5T^{2} \)
73 \( 1 - 578. iT - 3.89e5T^{2} \)
79 \( 1 - 745. iT - 4.93e5T^{2} \)
83 \( 1 + (286. + 286. i)T + 5.71e5iT^{2} \)
89 \( 1 + 203.T + 7.04e5T^{2} \)
97 \( 1 + 39.4T + 9.12e5T^{2} \)
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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

−12.32740174515637744112801375202, −11.47677690994016535539068127891, −10.33703218162823992274971964753, −9.256182203325393774498305398792, −8.171457830621322051356447457317, −6.84379305199001847590134142326, −6.03435072235121301567675044753, −4.65162704260025767840721117121, −3.52416972052946325366912015637, −0.934887860241167300571926652026, 0.30470618966050872620404116201, 3.12674004410195522960962647308, 4.08219583736876093610542403902, 5.74863999579112172186009206346, 6.83050119157820423072239467699, 7.34653947902691382672929682903, 9.237025175668660574171325132065, 10.14800136367253084901695814495, 10.95268108868262071195548342877, 11.97502126484810352287837644825

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