Properties

Degree $2$
Conductor $192$
Sign $0.281 + 0.959i$
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 + (6.49 − 6.49i)5-s − 3.94·7-s − 2.99i·9-s + (−4.31 − 4.31i)11-s + (4.06 + 4.06i)13-s − 15.9i·15-s − 14.5·17-s + (−4.94 + 4.94i)19-s + (−4.82 + 4.82i)21-s + 43.6·23-s − 59.3i·25-s + (−3.67 − 3.67i)27-s + (25.0 + 25.0i)29-s − 32.5i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.29 − 1.29i)5-s − 0.563·7-s − 0.333i·9-s + (−0.391 − 0.391i)11-s + (0.312 + 0.312i)13-s − 1.06i·15-s − 0.856·17-s + (−0.260 + 0.260i)19-s + (−0.229 + 0.229i)21-s + 1.89·23-s − 2.37i·25-s + (−0.136 − 0.136i)27-s + (0.865 + 0.865i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.281 + 0.959i$
Motivic weight: \(2\)
Character: $\chi_{192} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.281 + 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.53700 - 1.15105i\)
\(L(\frac12)\) \(\approx\) \(1.53700 - 1.15105i\)
\(L(2)\) 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 \( 1 + (-6.49 + 6.49i)T - 25iT^{2} \)
7 \( 1 + 3.94T + 49T^{2} \)
11 \( 1 + (4.31 + 4.31i)T + 121iT^{2} \)
13 \( 1 + (-4.06 - 4.06i)T + 169iT^{2} \)
17 \( 1 + 14.5T + 289T^{2} \)
19 \( 1 + (4.94 - 4.94i)T - 361iT^{2} \)
23 \( 1 - 43.6T + 529T^{2} \)
29 \( 1 + (-25.0 - 25.0i)T + 841iT^{2} \)
31 \( 1 + 32.5iT - 961T^{2} \)
37 \( 1 + (-4.14 + 4.14i)T - 1.36e3iT^{2} \)
41 \( 1 - 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (-16.1 - 16.1i)T + 1.84e3iT^{2} \)
47 \( 1 - 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (31.5 - 31.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-49.7 - 49.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-44.4 - 44.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 4.48e3iT^{2} \)
71 \( 1 + 24.1T + 5.04e3T^{2} \)
73 \( 1 - 10.7iT - 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (42.0 - 42.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−12.57071772515538938752435764402, −11.11491520683777241258590879333, −9.829455719576073885486384100018, −9.052373310879134518599876438926, −8.378876777825343970057715877569, −6.76171931294841219685140809075, −5.80179434342626663909549285909, −4.60922757661474778536108464651, −2.69574098828840375791067077269, −1.18613042558188598645488545250, 2.31202616218836651569536179449, 3.26794893808734076888771456868, 5.08153109677560647337735935074, 6.38585815801593950182441721997, 7.11418729049190911107942225822, 8.748678369470337614800569907287, 9.688873038063505281082268447302, 10.45267598002673430659356284420, 11.11495316961484342299443290905, 12.86676704976002921222711594319

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