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Properties

Label 2-192-24.11-c9-0-61
Degree $2$
Conductor $192$
Sign $-0.963 + 0.268i$
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  + (−120. + 71.3i)3-s − 275.·5-s − 3.90e3i·7-s + (9.50e3 − 1.72e4i)9-s − 8.18e4i·11-s − 1.43e5i·13-s + (3.33e4 − 1.96e4i)15-s − 2.75e5i·17-s + 4.06e5·19-s + (2.78e5 + 4.71e5i)21-s − 5.88e5·23-s − 1.87e6·25-s + (8.11e4 + 2.76e6i)27-s + 3.94e6·29-s − 2.49e6i·31-s + ⋯
L(s)  = 1  + (−0.861 + 0.508i)3-s − 0.197·5-s − 0.614i·7-s + (0.482 − 0.875i)9-s − 1.68i·11-s − 1.39i·13-s + (0.169 − 0.100i)15-s − 0.800i·17-s + 0.714·19-s + (0.312 + 0.528i)21-s − 0.438·23-s − 0.961·25-s + (0.0294 + 0.999i)27-s + 1.03·29-s − 0.485i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.963 + 0.268i$
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: \((2,\ 192,\ (\ :9/2),\ -0.963 + 0.268i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.007454887\)
\(L(\frac12)\) \(\approx\) \(1.007454887\)
\(L(\frac{11}{2})\) 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 \( 1 + (120. - 71.3i)T \)
good5 \( 1 + 275.T + 1.95e6T^{2} \)
7 \( 1 + 3.90e3iT - 4.03e7T^{2} \)
11 \( 1 + 8.18e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.43e5iT - 1.06e10T^{2} \)
17 \( 1 + 2.75e5iT - 1.18e11T^{2} \)
19 \( 1 - 4.06e5T + 3.22e11T^{2} \)
23 \( 1 + 5.88e5T + 1.80e12T^{2} \)
29 \( 1 - 3.94e6T + 1.45e13T^{2} \)
31 \( 1 + 2.49e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.06e7iT - 1.29e14T^{2} \)
41 \( 1 + 6.04e6iT - 3.27e14T^{2} \)
43 \( 1 - 1.19e7T + 5.02e14T^{2} \)
47 \( 1 + 6.34e7T + 1.11e15T^{2} \)
53 \( 1 - 1.03e8T + 3.29e15T^{2} \)
59 \( 1 - 1.08e8iT - 8.66e15T^{2} \)
61 \( 1 + 1.02e8iT - 1.16e16T^{2} \)
67 \( 1 + 1.21e8T + 2.72e16T^{2} \)
71 \( 1 - 2.05e8T + 4.58e16T^{2} \)
73 \( 1 + 3.12e6T + 5.88e16T^{2} \)
79 \( 1 - 2.61e8iT - 1.19e17T^{2} \)
83 \( 1 + 3.06e8iT - 1.86e17T^{2} \)
89 \( 1 + 2.28e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.06e9T + 7.60e17T^{2} \)
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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

−10.54790942044445186122889695355, −9.691317265926813938256710179189, −8.387483013906107302822349771059, −7.33606249188084826706292584584, −6.01030160987046811224189008971, −5.32846003619379887791681630436, −3.97568715675285563694556051579, −3.05083345597103021272587644616, −0.844112189246721131565198438217, −0.33617414600973374319608822964, 1.41706035112832771434493363202, 2.22902755581487718121599317653, 4.16962257574001933491720580589, 5.07488981432889690229772650221, 6.31932922736797385445788244914, 7.09154850195782187763929801079, 8.154483209784129481047503406389, 9.512551947291922222223097066956, 10.34382857911951841667333598377, 11.79040438408980482885160049351

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