Properties

Degree $2$
Conductor $192$
Sign $-0.577 + 0.816i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 2.44i)3-s − 2.82i·5-s − 10.3·7-s + (−2.99 − 8.48i)9-s − 14.6i·11-s + 6·13-s + (−6.92 − 4.89i)15-s + 22.6i·17-s − 10.3·19-s + (−18 + 25.4i)21-s − 29.3i·23-s + 17·25-s + (−25.9 − 7.34i)27-s − 31.1i·29-s + 31.1·31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s − 0.565i·5-s − 1.48·7-s + (−0.333 − 0.942i)9-s − 1.33i·11-s + 0.461·13-s + (−0.461 − 0.326i)15-s + 1.33i·17-s − 0.546·19-s + (−0.857 + 1.21i)21-s − 1.27i·23-s + 0.680·25-s + (−0.962 − 0.272i)27-s − 1.07i·29-s + 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.577 + 0.816i$
Motivic weight: \(2\)
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.613442 - 1.18507i\)
\(L(\frac12)\) \(\approx\) \(0.613442 - 1.18507i\)
\(L(2)\) 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 \( 1 + (-1.73 + 2.44i)T \)
good5 \( 1 + 2.82iT - 25T^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 - 6T + 169T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 + 31.1iT - 841T^{2} \)
31 \( 1 - 31.1T + 961T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 + 5.65iT - 1.68e3T^{2} \)
43 \( 1 + 10.3T + 1.84e3T^{2} \)
47 \( 1 - 58.7iT - 2.20e3T^{2} \)
53 \( 1 + 14.1iT - 2.80e3T^{2} \)
59 \( 1 + 14.6iT - 3.48e3T^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 - 114.T + 4.48e3T^{2} \)
71 \( 1 - 29.3iT - 5.04e3T^{2} \)
73 \( 1 + 30T + 5.32e3T^{2} \)
79 \( 1 - 31.1T + 6.24e3T^{2} \)
83 \( 1 - 73.4iT - 6.88e3T^{2} \)
89 \( 1 + 5.65iT - 7.92e3T^{2} \)
97 \( 1 - 90T + 9.40e3T^{2} \)
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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

−12.39358998129373484541994972047, −11.03472942988146290064782667668, −9.808908919737260454698142727809, −8.692018812970710695813805880800, −8.189674277927554316364104022505, −6.48029169248835227040527329569, −6.09999779994752604948496080480, −3.92772572483879986084655211127, −2.75157826694989290057115602843, −0.72401904427738249545463080342, 2.63146738052183941462554433015, 3.63447412582824223831344670747, 4.99712288124420771931009101710, 6.55217722705508720723459399003, 7.45436655754535338171488764119, 8.978208069888724654404750193979, 9.734421673167430953856171071532, 10.35446439392920892445786216936, 11.56247834361556696060022006960, 12.81754239359315597795646593055

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