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Properties

Label 2-192-48.11-c3-0-1
Degree $2$
Conductor $192$
Sign $-0.997 - 0.0715i$
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  + (4.81 + 1.96i)3-s + (−6.30 + 6.30i)5-s − 24.6·7-s + (19.3 + 18.8i)9-s + (−40.4 − 40.4i)11-s + (−47.3 + 47.3i)13-s + (−42.6 + 17.9i)15-s − 41.7i·17-s + (−10.6 − 10.6i)19-s + (−118. − 48.3i)21-s − 53.4i·23-s + 45.5i·25-s + (55.9 + 128. i)27-s + (105. + 105. i)29-s + 3.14i·31-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)3-s + (−0.563 + 0.563i)5-s − 1.33·7-s + (0.715 + 0.698i)9-s + (−1.10 − 1.10i)11-s + (−1.00 + 1.00i)13-s + (−0.734 + 0.309i)15-s − 0.595i·17-s + (−0.128 − 0.128i)19-s + (−1.23 − 0.502i)21-s − 0.484i·23-s + 0.364i·25-s + (0.398 + 0.917i)27-s + (0.674 + 0.674i)29-s + 0.0182i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0715i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.997 - 0.0715i$
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: \((2,\ 192,\ (\ :3/2),\ -0.997 - 0.0715i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0191377 + 0.534053i\)
\(L(\frac12)\) \(\approx\) \(0.0191377 + 0.534053i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.81 - 1.96i)T \)
good5 \( 1 + (6.30 - 6.30i)T - 125iT^{2} \)
7 \( 1 + 24.6T + 343T^{2} \)
11 \( 1 + (40.4 + 40.4i)T + 1.33e3iT^{2} \)
13 \( 1 + (47.3 - 47.3i)T - 2.19e3iT^{2} \)
17 \( 1 + 41.7iT - 4.91e3T^{2} \)
19 \( 1 + (10.6 + 10.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 53.4iT - 1.21e4T^{2} \)
29 \( 1 + (-105. - 105. i)T + 2.43e4iT^{2} \)
31 \( 1 - 3.14iT - 2.97e4T^{2} \)
37 \( 1 + (-42.1 - 42.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + (221. - 221. i)T - 7.95e4iT^{2} \)
47 \( 1 + 381.T + 1.03e5T^{2} \)
53 \( 1 + (294. - 294. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-445. - 445. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-21.8 + 21.8i)T - 2.26e5iT^{2} \)
67 \( 1 + (-572. - 572. i)T + 3.00e5iT^{2} \)
71 \( 1 + 612. iT - 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 + 427. iT - 4.93e5T^{2} \)
83 \( 1 + (245. - 245. i)T - 5.71e5iT^{2} \)
89 \( 1 - 188.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−12.76781162571590718993501957012, −11.48619701182015663154406345048, −10.39952757827362343147128243491, −9.614166159845775353960854315683, −8.610066720608453548360456835853, −7.48736278649870415722296838734, −6.58222168750440778548601783008, −4.86222179815134821519918971508, −3.38829092128614783085080445407, −2.70260768372896278763557226508, 0.19272898742004468144009693577, 2.37637919057443899084634983695, 3.56192744283485167093544957498, 4.96937863957389563379368784871, 6.61659308134903918506726080037, 7.67546534210656003458266138882, 8.363396176947150423315696723804, 9.799475594118248822935794758087, 10.10112184582191636408863946224, 12.05960366189308959771152786372

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