Properties

Label 2-192-12.11-c3-0-18
Degree $2$
Conductor $192$
Sign $-0.995 - 0.0957i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.497 − 5.17i)3-s + 18.4i·5-s − 17.1i·7-s + (−26.5 − 5.14i)9-s − 48.1·11-s − 33.7·13-s + (95.4 + 9.18i)15-s − 84.4i·17-s − 28.7i·19-s + (−88.7 − 8.54i)21-s − 96.4·23-s − 215.·25-s + (−39.8 + 134. i)27-s + 141. i·29-s + 51.8i·31-s + ⋯
L(s)  = 1  + (0.0957 − 0.995i)3-s + 1.65i·5-s − 0.926i·7-s + (−0.981 − 0.190i)9-s − 1.32·11-s − 0.719·13-s + (1.64 + 0.158i)15-s − 1.20i·17-s − 0.346i·19-s + (−0.922 − 0.0887i)21-s − 0.874·23-s − 1.72·25-s + (−0.283 + 0.958i)27-s + 0.904i·29-s + 0.300i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.995 - 0.0957i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.995 - 0.0957i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0116787 + 0.243291i\)
\(L(\frac12)\) \(\approx\) \(0.0116787 + 0.243291i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.497 + 5.17i)T \)
good5 \( 1 - 18.4iT - 125T^{2} \)
7 \( 1 + 17.1iT - 343T^{2} \)
11 \( 1 + 48.1T + 1.33e3T^{2} \)
13 \( 1 + 33.7T + 2.19e3T^{2} \)
17 \( 1 + 84.4iT - 4.91e3T^{2} \)
19 \( 1 + 28.7iT - 6.85e3T^{2} \)
23 \( 1 + 96.4T + 1.21e4T^{2} \)
29 \( 1 - 141. iT - 2.43e4T^{2} \)
31 \( 1 - 51.8iT - 2.97e4T^{2} \)
37 \( 1 + 323.T + 5.06e4T^{2} \)
41 \( 1 - 134. iT - 6.89e4T^{2} \)
43 \( 1 + 114. iT - 7.95e4T^{2} \)
47 \( 1 + 247.T + 1.03e5T^{2} \)
53 \( 1 + 169. iT - 1.48e5T^{2} \)
59 \( 1 - 605.T + 2.05e5T^{2} \)
61 \( 1 - 343.T + 2.26e5T^{2} \)
67 \( 1 + 900. iT - 3.00e5T^{2} \)
71 \( 1 - 331.T + 3.57e5T^{2} \)
73 \( 1 - 777.T + 3.89e5T^{2} \)
79 \( 1 - 587. iT - 4.93e5T^{2} \)
83 \( 1 + 1.46e3T + 5.71e5T^{2} \)
89 \( 1 - 78.6iT - 7.04e5T^{2} \)
97 \( 1 + 62.0T + 9.12e5T^{2} \)
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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

−11.47694867489683948017585753203, −10.68207973028739258698357112188, −9.860298910594785313128583878934, −8.115007750534033778685545296224, −7.17358319054141485738393188585, −6.84255895022695408003931774880, −5.29058200027412992245356459862, −3.28243183135704192652706482430, −2.32616750054547715979059021454, −0.096035897622916121670074040233, 2.25498465349939160197192251203, 4.05439831182903536930589407339, 5.17866339837514823949383744724, 5.72206459516361318917091615763, 8.085023567362313959246699669198, 8.543267597615957426630816888782, 9.605579801964053094819680272008, 10.36712728329172965854369229756, 11.77863467191008499537185555078, 12.51021041605447538497598532374

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