Properties

Label 2-192-192.179-c1-0-27
Degree $2$
Conductor $192$
Sign $-0.422 + 0.906i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.596 − 1.28i)2-s + (0.791 − 1.54i)3-s + (−1.28 − 1.52i)4-s + (0.236 + 1.18i)5-s + (−1.50 − 1.93i)6-s + (1.67 − 0.692i)7-s + (−2.73 + 0.738i)8-s + (−1.74 − 2.43i)9-s + (1.66 + 0.405i)10-s + (0.195 − 0.292i)11-s + (−3.37 + 0.773i)12-s + (1.51 + 0.302i)13-s + (0.109 − 2.55i)14-s + (2.01 + 0.575i)15-s + (−0.681 + 3.94i)16-s + (−5.35 + 5.35i)17-s + ⋯
L(s)  = 1  + (0.421 − 0.906i)2-s + (0.456 − 0.889i)3-s + (−0.644 − 0.764i)4-s + (0.105 + 0.530i)5-s + (−0.613 − 0.789i)6-s + (0.632 − 0.261i)7-s + (−0.965 + 0.261i)8-s + (−0.582 − 0.812i)9-s + (0.525 + 0.128i)10-s + (0.0590 − 0.0883i)11-s + (−0.974 + 0.223i)12-s + (0.421 + 0.0838i)13-s + (0.0292 − 0.683i)14-s + (0.520 + 0.148i)15-s + (−0.170 + 0.985i)16-s + (−1.29 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849951 - 1.33470i\)
\(L(\frac12)\) \(\approx\) \(0.849951 - 1.33470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.596 + 1.28i)T \)
3 \( 1 + (-0.791 + 1.54i)T \)
good5 \( 1 + (-0.236 - 1.18i)T + (-4.61 + 1.91i)T^{2} \)
7 \( 1 + (-1.67 + 0.692i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.195 + 0.292i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-1.51 - 0.302i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 + (5.35 - 5.35i)T - 17iT^{2} \)
19 \( 1 + (-0.449 - 0.0895i)T + (17.5 + 7.27i)T^{2} \)
23 \( 1 + (-5.61 - 2.32i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.38 - 1.59i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + (1.66 + 8.34i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.93 + 4.66i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.78 - 2.67i)T + (-16.4 - 39.7i)T^{2} \)
47 \( 1 + (-0.866 - 0.866i)T + 47iT^{2} \)
53 \( 1 + (-0.141 - 0.0944i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (8.97 - 1.78i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (11.0 - 7.39i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (2.80 + 4.19i)T + (-25.6 + 61.8i)T^{2} \)
71 \( 1 + (-2.24 - 5.41i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (6.08 - 14.6i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-5.00 - 5.00i)T + 79iT^{2} \)
83 \( 1 + (-1.64 + 8.26i)T + (-76.6 - 31.7i)T^{2} \)
89 \( 1 + (4.01 + 9.68i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 - 0.511iT - 97T^{2} \)
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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

−12.35394411540524837274900516852, −11.17697003096851622783598981666, −10.71022444933934316439945286453, −9.180901189654804134874753368467, −8.367179046261461463875424215042, −6.95993405506390460196103291172, −5.90443584524594786051086355042, −4.26619552546333218920755393537, −2.89871349168007790937850654588, −1.54368967724063531225195280967, 2.96449777513754953463166991760, 4.62820967241345295582557904058, 5.01465822191069229132087594183, 6.56150018159381236958657191531, 7.948458098962720203742017936811, 8.815927608593261015379463183548, 9.420077355225537170646030159852, 10.94077070410159251593432101521, 11.93705449734940757943761729955, 13.31488881317779207422370153556

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