Properties

Degree $2$
Conductor $192$
Sign $-0.850 - 0.526i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (1.00 + 1.00i)5-s − 10.0·7-s + 2.99i·9-s + (−2.26 + 2.26i)11-s + (−6.88 + 6.88i)13-s − 2.46i·15-s − 22.3·17-s + (16.8 + 16.8i)19-s + (12.2 + 12.2i)21-s − 33.2·23-s − 22.9i·25-s + (3.67 − 3.67i)27-s + (−24.6 + 24.6i)29-s + 41.3i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.201 + 0.201i)5-s − 1.43·7-s + 0.333i·9-s + (−0.205 + 0.205i)11-s + (−0.529 + 0.529i)13-s − 0.164i·15-s − 1.31·17-s + (0.889 + 0.889i)19-s + (0.584 + 0.584i)21-s − 1.44·23-s − 0.918i·25-s + (0.136 − 0.136i)27-s + (−0.849 + 0.849i)29-s + 1.33i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.850 - 0.526i$
Motivic weight: \(2\)
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.850 - 0.526i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0619680 + 0.217987i\)
\(L(\frac12)\) \(\approx\) \(0.0619680 + 0.217987i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (-1.00 - 1.00i)T + 25iT^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + (2.26 - 2.26i)T - 121iT^{2} \)
13 \( 1 + (6.88 - 6.88i)T - 169iT^{2} \)
17 \( 1 + 22.3T + 289T^{2} \)
19 \( 1 + (-16.8 - 16.8i)T + 361iT^{2} \)
23 \( 1 + 33.2T + 529T^{2} \)
29 \( 1 + (24.6 - 24.6i)T - 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (6.60 + 6.60i)T + 1.36e3iT^{2} \)
41 \( 1 + 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (-48.8 + 48.8i)T - 1.84e3iT^{2} \)
47 \( 1 + 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (-25.1 - 25.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (6.23 - 6.23i)T - 3.48e3iT^{2} \)
61 \( 1 + (-35.9 + 35.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (10.2 + 10.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 - 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−12.53628071663621548224580289593, −12.02383048136874815907009761014, −10.62277010935256653018552899142, −9.871436241690574431206786987171, −8.817497623601264720645769405405, −7.29044565321492026892003116323, −6.56131011202918394620695844487, −5.50955822357979717543339266825, −3.85205463757762357443987311277, −2.25064653277094551811092433716, 0.12757559482829528313080772147, 2.76575054861063589596480374997, 4.18892990415220500092817777205, 5.59553005898888879254599045687, 6.47892197104713462245987659196, 7.74295543233561163090524522651, 9.375751049131641705361382345865, 9.675718402126376413933540358132, 10.91679282712898733759423669317, 11.85254436048669285233096640657

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