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Properties

Label 2-192-48.35-c3-0-20
Degree $2$
Conductor $192$
Sign $-0.853 - 0.521i$
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  + (−0.563 − 5.16i)3-s + (−13.1 − 13.1i)5-s + 13.2·7-s + (−26.3 + 5.82i)9-s + (24.0 − 24.0i)11-s + (−30.7 − 30.7i)13-s + (−60.5 + 75.4i)15-s + 56.8i·17-s + (−74.2 + 74.2i)19-s + (−7.47 − 68.5i)21-s + 21.7i·23-s + 221. i·25-s + (44.9 + 132. i)27-s + (−102. + 102. i)29-s − 219. i·31-s + ⋯
L(s)  = 1  + (−0.108 − 0.994i)3-s + (−1.17 − 1.17i)5-s + 0.716·7-s + (−0.976 + 0.215i)9-s + (0.660 − 0.660i)11-s + (−0.656 − 0.656i)13-s + (−1.04 + 1.29i)15-s + 0.811i·17-s + (−0.896 + 0.896i)19-s + (−0.0777 − 0.712i)21-s + 0.197i·23-s + 1.77i·25-s + (0.320 + 0.947i)27-s + (−0.657 + 0.657i)29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.853 - 0.521i$
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: \((2,\ 192,\ (\ :3/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.168002 + 0.597060i\)
\(L(\frac12)\) \(\approx\) \(0.168002 + 0.597060i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.563 + 5.16i)T \)
good5 \( 1 + (13.1 + 13.1i)T + 125iT^{2} \)
7 \( 1 - 13.2T + 343T^{2} \)
11 \( 1 + (-24.0 + 24.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (30.7 + 30.7i)T + 2.19e3iT^{2} \)
17 \( 1 - 56.8iT - 4.91e3T^{2} \)
19 \( 1 + (74.2 - 74.2i)T - 6.85e3iT^{2} \)
23 \( 1 - 21.7iT - 1.21e4T^{2} \)
29 \( 1 + (102. - 102. i)T - 2.43e4iT^{2} \)
31 \( 1 + 219. iT - 2.97e4T^{2} \)
37 \( 1 + (-83.3 + 83.3i)T - 5.06e4iT^{2} \)
41 \( 1 - 7.92T + 6.89e4T^{2} \)
43 \( 1 + (-153. - 153. i)T + 7.95e4iT^{2} \)
47 \( 1 + 208.T + 1.03e5T^{2} \)
53 \( 1 + (390. + 390. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-221. + 221. i)T - 2.05e5iT^{2} \)
61 \( 1 + (416. + 416. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-284. + 284. i)T - 3.00e5iT^{2} \)
71 \( 1 + 26.9iT - 3.57e5T^{2} \)
73 \( 1 + 839. iT - 3.89e5T^{2} \)
79 \( 1 + 556. iT - 4.93e5T^{2} \)
83 \( 1 + (-400. - 400. i)T + 5.71e5iT^{2} \)
89 \( 1 - 974.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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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

−11.67412679798970651046260715182, −10.95371801899544094148679626472, −9.138512869492435047914908824679, −7.982147845654128812750600940494, −7.930307729731545987040389845790, −6.22994247121221820545477451196, −5.02615158201059719276011268724, −3.72949245816752778526146068435, −1.62238030511567074152953730436, −0.27835237280485727888423757551, 2.66735493395272996375506089644, 4.05309642209675957945199131260, 4.79386479370760758696444099924, 6.61242780528836624839842513652, 7.48525207203316029328670837841, 8.731758059945221637088660095631, 9.805432087943175245452540695706, 10.91380795496540325876327109424, 11.44151504249958689004030757458, 12.19235939631202270668137070614

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