Properties

Label 2-192-3.2-c10-0-75
Degree $2$
Conductor $192$
Sign $-0.996 - 0.0775i$
Analytic cond. $121.988$
Root an. cond. $11.0448$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (242. + 18.8i)3-s − 4.81e3i·5-s − 670.·7-s + (5.83e4 + 9.12e3i)9-s − 2.33e5i·11-s − 3.07e5·13-s + (9.07e4 − 1.16e6i)15-s − 6.72e5i·17-s − 1.55e6·19-s + (−1.62e5 − 1.26e4i)21-s + 5.57e6i·23-s − 1.34e7·25-s + (1.39e7 + 3.30e6i)27-s − 2.97e7i·29-s − 3.09e7·31-s + ⋯
L(s)  = 1  + (0.996 + 0.0775i)3-s − 1.54i·5-s − 0.0398·7-s + (0.987 + 0.154i)9-s − 1.44i·11-s − 0.828·13-s + (0.119 − 1.53i)15-s − 0.473i·17-s − 0.626·19-s + (−0.0397 − 0.00309i)21-s + 0.866i·23-s − 1.37·25-s + (0.973 + 0.230i)27-s − 1.44i·29-s − 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.996 - 0.0775i$
Analytic conductor: \(121.988\)
Root analytic conductor: \(11.0448\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :5),\ -0.996 - 0.0775i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.715235644\)
\(L(\frac12)\) \(\approx\) \(1.715235644\)
\(L(6)\) 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 \( 1 + (-242. - 18.8i)T \)
good5 \( 1 + 4.81e3iT - 9.76e6T^{2} \)
7 \( 1 + 670.T + 2.82e8T^{2} \)
11 \( 1 + 2.33e5iT - 2.59e10T^{2} \)
13 \( 1 + 3.07e5T + 1.37e11T^{2} \)
17 \( 1 + 6.72e5iT - 2.01e12T^{2} \)
19 \( 1 + 1.55e6T + 6.13e12T^{2} \)
23 \( 1 - 5.57e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.97e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.09e7T + 8.19e14T^{2} \)
37 \( 1 - 8.56e7T + 4.80e15T^{2} \)
41 \( 1 + 3.59e7iT - 1.34e16T^{2} \)
43 \( 1 + 3.66e7T + 2.16e16T^{2} \)
47 \( 1 - 3.28e7iT - 5.25e16T^{2} \)
53 \( 1 - 4.59e8iT - 1.74e17T^{2} \)
59 \( 1 - 4.88e8iT - 5.11e17T^{2} \)
61 \( 1 - 6.12e7T + 7.13e17T^{2} \)
67 \( 1 + 6.70e8T + 1.82e18T^{2} \)
71 \( 1 - 1.23e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.08e9T + 4.29e18T^{2} \)
79 \( 1 - 1.86e9T + 9.46e18T^{2} \)
83 \( 1 - 1.09e9iT - 1.55e19T^{2} \)
89 \( 1 + 5.19e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.07e10T + 7.37e19T^{2} \)
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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

−9.788099123918509381707135897720, −9.108594623325349083695238121938, −8.335438950634197297758227179348, −7.53953831555593046328922094208, −5.89600759491835282651997173530, −4.79268217449974299983473311373, −3.82187960124992596012363110338, −2.55061966285347703779829270537, −1.30267130746465844141741341327, −0.28036671028042017912498655810, 1.83467352148474514857011694273, 2.54219500353257052941683496170, 3.56908189253170827078516892934, 4.71803553128641788939446997497, 6.58101795147130437623267628498, 7.13355117650317986624490376361, 8.035789744778377786601266614632, 9.392070045300885325946210832207, 10.14876937507111928704398226845, 10.92147393489568962469108832153

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