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Properties

Label 2-192-24.11-c9-0-44
Degree $2$
Conductor $192$
Sign $0.175 - 0.984i$
Analytic cond. $98.8868$
Root an. cond. $9.94418$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (127. + 59.5i)3-s + 2.36e3·5-s + 8.20e3i·7-s + (1.25e4 + 1.51e4i)9-s + 3.90e4i·11-s − 1.27e5i·13-s + (3.00e5 + 1.40e5i)15-s + 8.03e4i·17-s + 6.27e5·19-s + (−4.88e5 + 1.04e6i)21-s + 1.35e6·23-s + 3.64e6·25-s + (6.97e5 + 2.67e6i)27-s − 5.39e6·29-s + 3.70e6i·31-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)3-s + 1.69·5-s + 1.29i·7-s + (0.639 + 0.768i)9-s + 0.803i·11-s − 1.23i·13-s + (1.53 + 0.718i)15-s + 0.233i·17-s + 1.10·19-s + (−0.548 + 1.17i)21-s + 1.00·23-s + 1.86·25-s + (0.252 + 0.967i)27-s − 1.41·29-s + 0.720i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(98.8868\)
Root analytic conductor: \(9.94418\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :9/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.822642942\)
\(L(\frac12)\) \(\approx\) \(4.822642942\)
\(L(\frac{11}{2})\) 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 \( 1 + (-127. - 59.5i)T \)
good5 \( 1 - 2.36e3T + 1.95e6T^{2} \)
7 \( 1 - 8.20e3iT - 4.03e7T^{2} \)
11 \( 1 - 3.90e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.27e5iT - 1.06e10T^{2} \)
17 \( 1 - 8.03e4iT - 1.18e11T^{2} \)
19 \( 1 - 6.27e5T + 3.22e11T^{2} \)
23 \( 1 - 1.35e6T + 1.80e12T^{2} \)
29 \( 1 + 5.39e6T + 1.45e13T^{2} \)
31 \( 1 - 3.70e6iT - 2.64e13T^{2} \)
37 \( 1 - 7.73e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.38e7iT - 3.27e14T^{2} \)
43 \( 1 + 2.57e7T + 5.02e14T^{2} \)
47 \( 1 - 1.68e7T + 1.11e15T^{2} \)
53 \( 1 - 9.27e7T + 3.29e15T^{2} \)
59 \( 1 + 1.04e8iT - 8.66e15T^{2} \)
61 \( 1 + 8.86e7iT - 1.16e16T^{2} \)
67 \( 1 + 2.67e8T + 2.72e16T^{2} \)
71 \( 1 - 2.94e8T + 4.58e16T^{2} \)
73 \( 1 + 1.46e8T + 5.88e16T^{2} \)
79 \( 1 - 3.47e7iT - 1.19e17T^{2} \)
83 \( 1 + 1.75e8iT - 1.86e17T^{2} \)
89 \( 1 - 9.34e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.35e9T + 7.60e17T^{2} \)
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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

−10.73055835416377459052344771899, −9.799722177218460087907170352304, −9.303993807998733788953789309854, −8.391003715104532165105058318409, −7.05493230289440399111008053409, −5.57925353328732473185786644613, −5.13501288448606725717190057951, −3.20281792582261067272948786428, −2.36503894159922474851987736699, −1.49862084432117804831275690105, 0.912563111682735616034514693128, 1.69363375034657950249033050710, 2.86255197500495004482433683936, 4.07311400145939881648377486045, 5.57871434021662603397058900384, 6.73410329032902291200444629102, 7.44325730678470266802775840991, 8.939982560832482248627381404166, 9.499246814278831193458504892429, 10.40847226360419175739838670460

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