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Properties

Label 2-192-192.131-c1-0-20
Degree $2$
Conductor $192$
Sign $0.937 - 0.349i$
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  + (1.35 + 0.412i)2-s + (1.67 − 0.455i)3-s + (1.66 + 1.11i)4-s + (−1.35 + 2.02i)5-s + (2.44 + 0.0730i)6-s + (−1.74 − 4.21i)7-s + (1.78 + 2.19i)8-s + (2.58 − 1.52i)9-s + (−2.65 + 2.17i)10-s + (−4.91 + 0.978i)11-s + (3.28 + 1.10i)12-s + (−2.86 + 1.91i)13-s + (−0.624 − 6.42i)14-s + (−1.33 + 3.99i)15-s + (1.51 + 3.70i)16-s + (1.70 − 1.70i)17-s + ⋯
L(s)  = 1  + (0.956 + 0.291i)2-s + (0.964 − 0.262i)3-s + (0.830 + 0.557i)4-s + (−0.603 + 0.903i)5-s + (0.999 + 0.0298i)6-s + (−0.659 − 1.59i)7-s + (0.631 + 0.775i)8-s + (0.861 − 0.507i)9-s + (−0.841 + 0.688i)10-s + (−1.48 + 0.295i)11-s + (0.947 + 0.319i)12-s + (−0.793 + 0.530i)13-s + (−0.166 − 1.71i)14-s + (−0.345 + 1.03i)15-s + (0.378 + 0.925i)16-s + (0.413 − 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.937 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19613 + 0.395886i\)
\(L(\frac12)\) \(\approx\) \(2.19613 + 0.395886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.412i)T \)
3 \( 1 + (-1.67 + 0.455i)T \)
good5 \( 1 + (1.35 - 2.02i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (1.74 + 4.21i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.91 - 0.978i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (2.86 - 1.91i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-1.70 + 1.70i)T - 17iT^{2} \)
19 \( 1 + (-2.92 + 1.95i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.396 + 0.956i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.283 + 1.42i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 1.28T + 31T^{2} \)
37 \( 1 + (4.96 - 7.42i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.448 - 0.185i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.10 + 0.418i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (3.15 + 3.15i)T + 47iT^{2} \)
53 \( 1 + (0.610 + 3.06i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-12.0 - 8.03i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-1.00 + 5.06i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (14.2 + 2.84i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-4.14 + 1.71i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.01 - 1.24i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-8.12 - 8.12i)T + 79iT^{2} \)
83 \( 1 + (1.43 + 2.14i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (3.57 - 1.48i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 6.01iT - 97T^{2} \)
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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

−12.99234781427708536727938033952, −11.84771240599027258370745061356, −10.61378533633786447681436349510, −9.887477234453341484822411638855, −7.962766652691927635750217756665, −7.25082529680647023584990920574, −6.84736167860980705238995951190, −4.77838469901570388370115838160, −3.56994338505546077384466085038, −2.73033977771521570872410459896, 2.43110405570636578633746369240, 3.38251205208159666266715416267, 4.94531830285075797864254792413, 5.66561873576203058965758989621, 7.54027886722094201656052018630, 8.431441535287354732877013058998, 9.544876442021731595862072496602, 10.48857486028379797459757546949, 12.00392009530336509086460210102, 12.62714243039694072092523757567

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