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Properties

Label 2-192-12.11-c3-0-13
Degree $2$
Conductor $192$
Sign $0.0277 + 0.999i$
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  + (−5.19 + 0.143i)3-s − 7.96i·5-s + 25.6i·7-s + (26.9 − 1.49i)9-s + 23.3·11-s − 55.6·13-s + (1.14 + 41.3i)15-s − 115. i·17-s − 82.4i·19-s + (−3.69 − 133. i)21-s − 28.1·23-s + 61.5·25-s + (−139. + 11.6i)27-s − 105. i·29-s − 226. i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0277i)3-s − 0.712i·5-s + 1.38i·7-s + (0.998 − 0.0553i)9-s + 0.639·11-s − 1.18·13-s + (0.0197 + 0.712i)15-s − 1.64i·17-s − 0.995i·19-s + (−0.0384 − 1.38i)21-s − 0.255·23-s + 0.492·25-s + (−0.996 + 0.0830i)27-s − 0.676i·29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0277 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0277 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.0277 + 0.999i$
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: \((2,\ 192,\ (\ :3/2),\ 0.0277 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.629905 - 0.612686i\)
\(L(\frac12)\) \(\approx\) \(0.629905 - 0.612686i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.19 - 0.143i)T \)
good5 \( 1 + 7.96iT - 125T^{2} \)
7 \( 1 - 25.6iT - 343T^{2} \)
11 \( 1 - 23.3T + 1.33e3T^{2} \)
13 \( 1 + 55.6T + 2.19e3T^{2} \)
17 \( 1 + 115. iT - 4.91e3T^{2} \)
19 \( 1 + 82.4iT - 6.85e3T^{2} \)
23 \( 1 + 28.1T + 1.21e4T^{2} \)
29 \( 1 + 105. iT - 2.43e4T^{2} \)
31 \( 1 + 226. iT - 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 + 446. iT - 6.89e4T^{2} \)
43 \( 1 - 90.6iT - 7.95e4T^{2} \)
47 \( 1 + 446.T + 1.03e5T^{2} \)
53 \( 1 - 332. iT - 1.48e5T^{2} \)
59 \( 1 + 261.T + 2.05e5T^{2} \)
61 \( 1 + 232.T + 2.26e5T^{2} \)
67 \( 1 + 636. iT - 3.00e5T^{2} \)
71 \( 1 - 449.T + 3.57e5T^{2} \)
73 \( 1 - 354.T + 3.89e5T^{2} \)
79 \( 1 + 368. iT - 4.93e5T^{2} \)
83 \( 1 - 913.T + 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 - 521.T + 9.12e5T^{2} \)
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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

−11.94979168107613166567575482134, −11.19883283314074128275825234398, −9.551685774756188659792226718916, −9.220791059575593089342620603949, −7.64062320922187194451014773914, −6.43359474033067766237022964521, −5.30256326097154144490672035974, −4.61516872086037179307663908899, −2.41437188504169083477318887591, −0.47286106827891498172927911098, 1.37793048718440429698592870252, 3.64079878502229648477803470570, 4.73183911990799184156090358195, 6.27095606530347154362563433373, 6.95859791635937675971813579253, 7.967956135880004375622397836542, 9.848472189731414245239274148456, 10.42610168677986121656296973142, 11.17715104763268201377594658197, 12.29284676661516152296988693090

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