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Properties

Label 2-192-64.29-c1-0-5
Degree $2$
Conductor $192$
Sign $0.403 - 0.915i$
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  + (1.32 + 0.481i)2-s + (−0.980 − 0.195i)3-s + (1.53 + 1.28i)4-s + (−1.96 + 2.94i)5-s + (−1.21 − 0.731i)6-s + (2.86 − 1.18i)7-s + (1.42 + 2.44i)8-s + (0.923 + 0.382i)9-s + (−4.03 + 2.96i)10-s + (0.555 + 2.79i)11-s + (−1.25 − 1.55i)12-s + (−2.33 − 3.49i)13-s + (4.38 − 0.199i)14-s + (2.50 − 2.50i)15-s + (0.722 + 3.93i)16-s + (−0.737 − 0.737i)17-s + ⋯
L(s)  = 1  + (0.940 + 0.340i)2-s + (−0.566 − 0.112i)3-s + (0.768 + 0.640i)4-s + (−0.879 + 1.31i)5-s + (−0.494 − 0.298i)6-s + (1.08 − 0.448i)7-s + (0.504 + 0.863i)8-s + (0.307 + 0.127i)9-s + (−1.27 + 0.937i)10-s + (0.167 + 0.842i)11-s + (−0.362 − 0.448i)12-s + (−0.648 − 0.970i)13-s + (1.17 − 0.0531i)14-s + (0.646 − 0.646i)15-s + (0.180 + 0.983i)16-s + (−0.178 − 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.403 - 0.915i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.403 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36459 + 0.890032i\)
\(L(\frac12)\) \(\approx\) \(1.36459 + 0.890032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.481i)T \)
3 \( 1 + (0.980 + 0.195i)T \)
good5 \( 1 + (1.96 - 2.94i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-2.86 + 1.18i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.555 - 2.79i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (2.33 + 3.49i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (0.737 + 0.737i)T + 17iT^{2} \)
19 \( 1 + (-1.47 + 0.982i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-2.86 + 6.91i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.59 + 8.00i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 4.07iT - 31T^{2} \)
37 \( 1 + (6.70 + 4.48i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (2.49 - 6.01i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-5.42 + 1.07i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (4.06 + 4.06i)T + 47iT^{2} \)
53 \( 1 + (-2.59 - 13.0i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-3.86 + 5.78i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.22 - 0.242i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-10.0 - 2.00i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (7.13 - 2.95i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.12 + 0.467i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-5.29 + 5.29i)T - 79iT^{2} \)
83 \( 1 + (-1.11 + 0.747i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-0.767 - 1.85i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−12.53850637893659738011039540861, −11.81259773055897686458135276205, −10.97455834692525709470586924072, −10.35834589438241317762919803720, −8.067920564213780595927870279331, −7.33631724568450472409774920101, −6.64050562417473513550246843090, −5.09553398095577510807290318637, −4.16596676778936388802823221354, −2.64039418273949431928822405494, 1.46280015116527458720937713379, 3.77683995384644278106160750882, 4.90610778735333990362930056658, 5.42174227942348188926033591690, 7.05452578053843407247237793136, 8.317987098375507475367480210950, 9.388707565850117287948877581881, 10.97197269502747037126649073975, 11.70272547668905488209702321478, 12.07707069500113767006296916224

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