Properties

Label 2-192-192.107-c1-0-20
Degree $2$
Conductor $192$
Sign $-0.973 + 0.230i$
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  + (−0.890 − 1.09i)2-s + (−0.706 + 1.58i)3-s + (−0.412 + 1.95i)4-s + (−1.50 − 2.24i)5-s + (2.36 − 0.632i)6-s + (−0.0860 + 0.207i)7-s + (2.51 − 1.29i)8-s + (−2.00 − 2.23i)9-s + (−1.12 + 3.64i)10-s + (−3.68 − 0.732i)11-s + (−2.80 − 2.03i)12-s + (−1.53 − 1.02i)13-s + (0.304 − 0.0905i)14-s + (4.61 − 0.786i)15-s + (−3.65 − 1.61i)16-s + (−4.44 − 4.44i)17-s + ⋯
L(s)  = 1  + (−0.629 − 0.776i)2-s + (−0.407 + 0.913i)3-s + (−0.206 + 0.978i)4-s + (−0.671 − 1.00i)5-s + (0.966 − 0.258i)6-s + (−0.0325 + 0.0784i)7-s + (0.889 − 0.456i)8-s + (−0.667 − 0.744i)9-s + (−0.357 + 1.15i)10-s + (−1.11 − 0.220i)11-s + (−0.809 − 0.587i)12-s + (−0.425 − 0.284i)13-s + (0.0814 − 0.0241i)14-s + (1.19 − 0.203i)15-s + (−0.914 − 0.403i)16-s + (−1.07 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0250747 - 0.214460i\)
\(L(\frac12)\) \(\approx\) \(0.0250747 - 0.214460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.890 + 1.09i)T \)
3 \( 1 + (0.706 - 1.58i)T \)
good5 \( 1 + (1.50 + 2.24i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (0.0860 - 0.207i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (3.68 + 0.732i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.53 + 1.02i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (4.44 + 4.44i)T + 17iT^{2} \)
19 \( 1 + (3.66 + 2.45i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.88 - 4.55i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.40 - 7.06i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 6.44T + 31T^{2} \)
37 \( 1 + (5.00 + 7.48i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.55 - 1.88i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.54 + 0.307i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-4.26 + 4.26i)T - 47iT^{2} \)
53 \( 1 + (-1.02 + 5.13i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (7.47 - 4.99i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.815 - 4.10i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (8.05 - 1.60i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (2.52 + 1.04i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.90 + 2.44i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (8.79 - 8.79i)T - 79iT^{2} \)
83 \( 1 + (-4.93 + 7.39i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (5.44 + 2.25i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−11.89055340975023631444061061524, −11.02829895238887624959338505048, −10.24023815507320519502970201473, −9.045250093697862099699704687029, −8.551969183134783222862379494783, −7.21337380455634352954950177316, −5.15804814752536256079417404517, −4.40442507516008734045864364101, −2.90828926601366932020056242272, −0.23226156912467906521001749464, 2.32681323039303958885338750722, 4.64847231265795395719913951492, 6.20645845484324922517170372890, 6.83295406289217811464806469227, 7.83386497530344149601316749703, 8.497720665794850772904675541027, 10.34363907192515225074113343237, 10.75286912530003536749563414214, 11.91472229354009700966733146326, 13.09917349198933536146872533741

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