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Properties

Label 2-192-64.61-c1-0-7
Degree $2$
Conductor $192$
Sign $0.518 + 0.855i$
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  + (−0.796 − 1.16i)2-s + (0.195 − 0.980i)3-s + (−0.731 + 1.86i)4-s + (2.41 + 1.61i)5-s + (−1.30 + 0.552i)6-s + (1.09 − 0.453i)7-s + (2.75 − 0.626i)8-s + (−0.923 − 0.382i)9-s + (−0.0370 − 4.10i)10-s + (5.11 − 1.01i)11-s + (1.68 + 1.08i)12-s + (−3.73 + 2.49i)13-s + (−1.40 − 0.918i)14-s + (2.05 − 2.05i)15-s + (−2.92 − 2.72i)16-s + (−0.516 − 0.516i)17-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (0.112 − 0.566i)3-s + (−0.365 + 0.930i)4-s + (1.08 + 0.721i)5-s + (−0.531 + 0.225i)6-s + (0.413 − 0.171i)7-s + (0.975 − 0.221i)8-s + (−0.307 − 0.127i)9-s + (−0.0117 − 1.29i)10-s + (1.54 − 0.307i)11-s + (0.485 + 0.312i)12-s + (−1.03 + 0.691i)13-s + (−0.374 − 0.245i)14-s + (0.530 − 0.530i)15-s + (−0.732 − 0.681i)16-s + (−0.125 − 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.518 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.963876 - 0.542888i\)
\(L(\frac12)\) \(\approx\) \(0.963876 - 0.542888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.796 + 1.16i)T \)
3 \( 1 + (-0.195 + 0.980i)T \)
good5 \( 1 + (-2.41 - 1.61i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-1.09 + 0.453i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-5.11 + 1.01i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (3.73 - 2.49i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (0.516 + 0.516i)T + 17iT^{2} \)
19 \( 1 + (2.92 + 4.37i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.18 + 2.86i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.523 - 0.104i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 - 9.33iT - 31T^{2} \)
37 \( 1 + (-3.58 + 5.36i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (3.54 - 8.55i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.828 + 4.16i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (1.07 + 1.07i)T + 47iT^{2} \)
53 \( 1 + (-3.79 + 0.755i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (10.0 + 6.68i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (2.46 - 12.4i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (0.394 - 1.98i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (0.0704 - 0.0291i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (10.6 + 4.41i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.83 - 3.83i)T - 79iT^{2} \)
83 \( 1 + (8.97 + 13.4i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-5.65 - 13.6i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 3.93iT - 97T^{2} \)
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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

−12.18867388059340477315729495482, −11.37478849614242775126234792428, −10.45145505568239431101727003241, −9.388713496824247737414737394606, −8.705373652355917763106805084631, −7.15987436286678203096870438798, −6.48684629225925824967571644069, −4.55092268824924031390396828459, −2.82313293274660489503861077247, −1.64468602105439918239714844021, 1.73500156940972957911306170401, 4.35767671047728870129165504216, 5.41770806464380857291342937061, 6.31687357261219420817674314293, 7.77883671091058324282524205894, 8.865044759931703432775889755262, 9.583972802315658147952029603773, 10.19797950328329166640081918993, 11.60993242036417199631338248059, 12.87561477743454787314228353315

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