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Properties

Label 2-192-16.5-c3-0-10
Degree $2$
Conductor $192$
Sign $-0.956 - 0.291i$
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 + (−8.83 − 8.83i)5-s + 29.4i·7-s − 8.99i·9-s + (−44.6 − 44.6i)11-s + (6.83 − 6.83i)13-s − 37.4·15-s − 56.8·17-s + (−91.0 + 91.0i)19-s + (62.5 + 62.5i)21-s + 96.8i·23-s + 31.0i·25-s + (−19.0 − 19.0i)27-s + (−59.2 + 59.2i)29-s − 103.·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.790 − 0.790i)5-s + 1.59i·7-s − 0.333i·9-s + (−1.22 − 1.22i)11-s + (0.145 − 0.145i)13-s − 0.645·15-s − 0.810·17-s + (−1.09 + 1.09i)19-s + (0.649 + 0.649i)21-s + 0.877i·23-s + 0.248i·25-s + (−0.136 − 0.136i)27-s + (−0.379 + 0.379i)29-s − 0.597·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.956 - 0.291i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.956 - 0.291i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0127330 + 0.0855864i\)
\(L(\frac12)\) \(\approx\) \(0.0127330 + 0.0855864i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.83 + 8.83i)T + 125iT^{2} \)
7 \( 1 - 29.4iT - 343T^{2} \)
11 \( 1 + (44.6 + 44.6i)T + 1.33e3iT^{2} \)
13 \( 1 + (-6.83 + 6.83i)T - 2.19e3iT^{2} \)
17 \( 1 + 56.8T + 4.91e3T^{2} \)
19 \( 1 + (91.0 - 91.0i)T - 6.85e3iT^{2} \)
23 \( 1 - 96.8iT - 1.21e4T^{2} \)
29 \( 1 + (59.2 - 59.2i)T - 2.43e4iT^{2} \)
31 \( 1 + 103.T + 2.97e4T^{2} \)
37 \( 1 + (79.5 + 79.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 105. iT - 6.89e4T^{2} \)
43 \( 1 + (39.9 + 39.9i)T + 7.95e4iT^{2} \)
47 \( 1 + 9.34T + 1.03e5T^{2} \)
53 \( 1 + (-245. - 245. i)T + 1.48e5iT^{2} \)
59 \( 1 + (345. + 345. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-370. + 370. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-595. + 595. i)T - 3.00e5iT^{2} \)
71 \( 1 + 493. iT - 3.57e5T^{2} \)
73 \( 1 - 33.2iT - 3.89e5T^{2} \)
79 \( 1 + 552.T + 4.93e5T^{2} \)
83 \( 1 + (-18.5 + 18.5i)T - 5.71e5iT^{2} \)
89 \( 1 + 934. iT - 7.04e5T^{2} \)
97 \( 1 - 1.42e3T + 9.12e5T^{2} \)
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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

−11.74656565064498365132387286031, −10.73836122530850237615318867951, −9.089005065636046204733882410418, −8.481566360072555109209359258006, −7.83640261176103109412063682038, −6.10052515193482270914587771996, −5.19511465182600795042067590673, −3.53503796331434230601710501604, −2.13826331194557060128282794851, −0.03349257335016548783261769603, 2.48952740332799377970086973787, 3.95652990876764023409905343830, 4.68713191402931956896338944821, 6.87712527451488753992320151188, 7.36340952673530496542328805863, 8.463735378709410861602427297616, 9.944123694343245253455553915067, 10.65600563866759084890619908101, 11.24796827790458112527647587143, 12.86998805680157883781033973021

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