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Properties

Label 2-192-48.11-c3-0-21
Degree $2$
Conductor $192$
Sign $-0.977 + 0.209i$
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  + (1.13 − 5.07i)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 + (−34.2 + 153. i)21-s − 1.28i·23-s − 128. i·25-s + (−86.1 + 110. i)27-s + (−119. − 119. i)29-s − 172. i·31-s + ⋯
L(s)  = 1  + (0.218 − 0.975i)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 + (−0.356 + 1.59i)21-s − 0.0116i·23-s − 1.03i·25-s + (−0.613 + 0.789i)27-s + (−0.764 − 0.764i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.977 + 0.209i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.977 + 0.209i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.129705 - 1.22748i\)
\(L(\frac12)\) \(\approx\) \(0.129705 - 1.22748i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.13 + 5.07i)T \)
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

−12.05240776312709683201073055778, −10.46631744658389913093252237018, −9.431066326688607696263515881993, −8.731689376711612369128592887734, −7.49361528855878113291447167833, −6.09138917966828754628929181716, −5.72477936913770669131032118875, −3.54199990024611346731593196235, −2.06724172410520033130698812096, −0.49422898274047715067767349067, 2.63451590091626989049489521286, 3.38995338464247217162828905773, 5.13305485383914127388566338380, 6.27835648753290152339272665996, 7.17176049687126827947838044965, 9.015880737073113600511969638480, 9.728882997159512336713727199861, 10.27616417742597475962935068946, 11.24217201471197216024476272226, 12.69070371964135040818707633288

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