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Properties

Label 2-192-16.13-c3-0-3
Degree $2$
Conductor $192$
Sign $0.314 - 0.949i$
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  + (−2.12 − 2.12i)3-s + (0.644 − 0.644i)5-s + 7.13i·7-s + 8.99i·9-s + (−25.4 + 25.4i)11-s + (−14.6 − 14.6i)13-s − 2.73·15-s + 71.4·17-s + (43.6 + 43.6i)19-s + (15.1 − 15.1i)21-s + 211. i·23-s + 124. i·25-s + (19.0 − 19.0i)27-s + (5.84 + 5.84i)29-s + 107.·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.0576 − 0.0576i)5-s + 0.385i·7-s + 0.333i·9-s + (−0.697 + 0.697i)11-s + (−0.311 − 0.311i)13-s − 0.0470·15-s + 1.01·17-s + (0.526 + 0.526i)19-s + (0.157 − 0.157i)21-s + 1.92i·23-s + 0.993i·25-s + (0.136 − 0.136i)27-s + (0.0374 + 0.0374i)29-s + 0.624·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.922574 + 0.665907i\)
\(L(\frac12)\) \(\approx\) \(0.922574 + 0.665907i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-0.644 + 0.644i)T - 125iT^{2} \)
7 \( 1 - 7.13iT - 343T^{2} \)
11 \( 1 + (25.4 - 25.4i)T - 1.33e3iT^{2} \)
13 \( 1 + (14.6 + 14.6i)T + 2.19e3iT^{2} \)
17 \( 1 - 71.4T + 4.91e3T^{2} \)
19 \( 1 + (-43.6 - 43.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 211. iT - 1.21e4T^{2} \)
29 \( 1 + (-5.84 - 5.84i)T + 2.43e4iT^{2} \)
31 \( 1 - 107.T + 2.97e4T^{2} \)
37 \( 1 + (184. - 184. i)T - 5.06e4iT^{2} \)
41 \( 1 - 360. iT - 6.89e4T^{2} \)
43 \( 1 + (-312. + 312. i)T - 7.95e4iT^{2} \)
47 \( 1 + 343.T + 1.03e5T^{2} \)
53 \( 1 + (249. - 249. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-152. + 152. i)T - 2.05e5iT^{2} \)
61 \( 1 + (525. + 525. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-35.3 - 35.3i)T + 3.00e5iT^{2} \)
71 \( 1 - 784. iT - 3.57e5T^{2} \)
73 \( 1 + 800. iT - 3.89e5T^{2} \)
79 \( 1 + 548.T + 4.93e5T^{2} \)
83 \( 1 + (464. + 464. i)T + 5.71e5iT^{2} \)
89 \( 1 - 302. iT - 7.04e5T^{2} \)
97 \( 1 - 1.56e3T + 9.12e5T^{2} \)
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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

−12.24479878876400809421750571757, −11.50481828466269170101975925146, −10.24751956505260997845720717547, −9.465361570219364520380368919478, −7.956777033202954578681741504947, −7.29040242543020173879224193883, −5.80296271239984874488595481578, −5.04117699935209022974277484641, −3.18910523481190143166755573875, −1.51955599430505472131232086502, 0.54894492285560181882384901147, 2.79536713589259206095256410908, 4.29343954441764035774558698843, 5.43298559566854467645707587753, 6.57798553543347591232901507854, 7.81111136159289286703201788860, 8.930641505627836044696754813323, 10.17961437388251237283721593469, 10.71535076082763610046606781155, 11.86864092277221911316692182096

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