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Properties

Label 2-192-24.5-c2-0-4
Degree $2$
Conductor $192$
Sign $0.547 - 0.836i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 2i)3-s + 7.74·5-s − 3.46·7-s + (1.00 − 8.94i)9-s + 13.4·11-s + 20.7i·13-s + (−17.3 + 15.4i)15-s + 4i·19-s + (7.74 − 6.92i)21-s + 30.9i·23-s + 35.0·25-s + (15.6 + 22.0i)27-s + 7.74·29-s + 24.2·31-s + (−30.0 + 26.8i)33-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)3-s + 1.54·5-s − 0.494·7-s + (0.111 − 0.993i)9-s + 1.21·11-s + 1.59i·13-s + (−1.15 + 1.03i)15-s + 0.210i·19-s + (0.368 − 0.329i)21-s + 1.34i·23-s + 1.40·25-s + (0.579 + 0.814i)27-s + 0.267·29-s + 0.782·31-s + (−0.909 + 0.813i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.547 - 0.836i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.547 - 0.836i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32469 + 0.716413i\)
\(L(\frac12)\) \(\approx\) \(1.32469 + 0.716413i\)
\(L(2)\) 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 \( 1 + (2.23 - 2i)T \)
good5 \( 1 - 7.74T + 25T^{2} \)
7 \( 1 + 3.46T + 49T^{2} \)
11 \( 1 - 13.4T + 121T^{2} \)
13 \( 1 - 20.7iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 4iT - 361T^{2} \)
23 \( 1 - 30.9iT - 529T^{2} \)
29 \( 1 - 7.74T + 841T^{2} \)
31 \( 1 - 24.2T + 961T^{2} \)
37 \( 1 - 34.6iT - 1.36e3T^{2} \)
41 \( 1 + 53.6iT - 1.68e3T^{2} \)
43 \( 1 + 52iT - 1.84e3T^{2} \)
47 \( 1 + 61.9iT - 2.20e3T^{2} \)
53 \( 1 + 54.2T + 2.80e3T^{2} \)
59 \( 1 + 40.2T + 3.48e3T^{2} \)
61 \( 1 + 6.92iT - 3.72e3T^{2} \)
67 \( 1 + 28iT - 4.48e3T^{2} \)
71 \( 1 + 30.9iT - 5.04e3T^{2} \)
73 \( 1 - 74T + 5.32e3T^{2} \)
79 \( 1 - 51.9T + 6.24e3T^{2} \)
83 \( 1 - 120.T + 6.88e3T^{2} \)
89 \( 1 + 53.6iT - 7.92e3T^{2} \)
97 \( 1 + 62T + 9.40e3T^{2} \)
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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

−12.24437324879874466688014930792, −11.51875206955923418813755064803, −10.30651762933068507630478944916, −9.486203028078261203059203845890, −9.073634531471297601283750870148, −6.73340421515629079462038482446, −6.24639711622908987636033063325, −5.08033902783299316075689974768, −3.72867524643368045531886266905, −1.66792950959333713365548033285, 1.10903924047594198540369993864, 2.69993230528348335204375209185, 4.90100494894744230906227999652, 6.16120515584086598907405219719, 6.44883910308677091114585550712, 8.006157836152988078581675464375, 9.371521302266763424410531888656, 10.21017686158976632884889603592, 11.10071098071083658750992960595, 12.47404905761009406903047048379

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