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Properties

Label 2-192-12.11-c3-0-2
Degree $2$
Conductor $192$
Sign $-i$
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  − 5.19i·3-s + 31.1i·7-s − 27·9-s − 70·13-s + 155. i·19-s + 162·21-s + 125·25-s + 140. i·27-s + 155. i·31-s − 110·37-s + 363. i·39-s − 218. i·43-s − 629·49-s + 810·57-s − 182·61-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.68i·7-s − 9-s − 1.49·13-s + 1.88i·19-s + 1.68·21-s + 25-s + 1.00i·27-s + 0.903i·31-s − 0.488·37-s + 1.49i·39-s − 0.773i·43-s − 1.83·49-s + 1.88·57-s − 0.382·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-i$
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: \((2,\ 192,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.649630 + 0.649630i\)
\(L(\frac12)\) \(\approx\) \(0.649630 + 0.649630i\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.19iT \)
good5 \( 1 - 125T^{2} \)
7 \( 1 - 31.1iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 70T + 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 155. iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + 110T + 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 218. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 182T + 2.26e5T^{2} \)
67 \( 1 - 654. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 1.19e3T + 3.89e5T^{2} \)
79 \( 1 + 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−12.11404556454237517898715750579, −11.97080426651678527172442316758, −10.35749313763207394316265647467, −9.103640294605888181883488790002, −8.288385667591260325808733910968, −7.21140623894425760295039773128, −6.01083526005726888950259177745, −5.15931570002782166839203250713, −2.94909743387969823483662727294, −1.84805953450394109073391430639, 0.38550567449165372010776371804, 2.88471476683056850439371472394, 4.29923251272244425073223636634, 4.99119024932835093425875404787, 6.75966867381010102619316302029, 7.66425583564609232077092167683, 9.090726508523707183617641411658, 9.959624652498773104774836292666, 10.72874747690761892007084898356, 11.52925075223636367977274557246

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