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Properties

Label 2-192-3.2-c2-0-9
Degree $2$
Conductor $192$
Sign $0.881 + 0.471i$
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  + (2.64 + 1.41i)3-s − 7.48i·5-s + 5.29·7-s + (5 + 7.48i)9-s − 14.1i·11-s − 10·13-s + (10.5 − 19.7i)15-s + 26.4·19-s + (14.0 + 7.48i)21-s + 16.9i·23-s − 31·25-s + (2.64 + 26.8i)27-s + 37.4i·29-s + 26.4·31-s + (20.0 − 37.4i)33-s + ⋯
L(s)  = 1  + (0.881 + 0.471i)3-s − 1.49i·5-s + 0.755·7-s + (0.555 + 0.831i)9-s − 1.28i·11-s − 0.769·13-s + (0.705 − 1.31i)15-s + 1.39·19-s + (0.666 + 0.356i)21-s + 0.737i·23-s − 1.23·25-s + (0.0979 + 0.995i)27-s + 1.29i·29-s + 0.853·31-s + (0.606 − 1.13i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.04190 - 0.511480i\)
\(L(\frac12)\) \(\approx\) \(2.04190 - 0.511480i\)
\(L(2)\) 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 \( 1 + (-2.64 - 1.41i)T \)
good5 \( 1 + 7.48iT - 25T^{2} \)
7 \( 1 - 5.29T + 49T^{2} \)
11 \( 1 + 14.1iT - 121T^{2} \)
13 \( 1 + 10T + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 26.4T + 361T^{2} \)
23 \( 1 - 16.9iT - 529T^{2} \)
29 \( 1 - 37.4iT - 841T^{2} \)
31 \( 1 - 26.4T + 961T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 + 14.9iT - 1.68e3T^{2} \)
43 \( 1 + 58.2T + 1.84e3T^{2} \)
47 \( 1 - 11.3iT - 2.20e3T^{2} \)
53 \( 1 + 37.4iT - 2.80e3T^{2} \)
59 \( 1 - 98.9iT - 3.48e3T^{2} \)
61 \( 1 + 90T + 3.72e3T^{2} \)
67 \( 1 - 5.29T + 4.48e3T^{2} \)
71 \( 1 - 28.2iT - 5.04e3T^{2} \)
73 \( 1 + 30T + 5.32e3T^{2} \)
79 \( 1 - 26.4T + 6.24e3T^{2} \)
83 \( 1 - 25.4iT - 6.88e3T^{2} \)
89 \( 1 + 74.8iT - 7.92e3T^{2} \)
97 \( 1 - 10T + 9.40e3T^{2} \)
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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

−12.26416020115277396760895508420, −11.32298306220038926558118037515, −10.00789617653731102628542775530, −9.030268822524644012880979047626, −8.410912422517281773935282916688, −7.51621170320182060498473890205, −5.40499065924552829693748214946, −4.72283340238357699330839656954, −3.26774967706568332129687921590, −1.36249060372575857978381228555, 2.02920292395017213464743587191, 3.09256723396046892192849307120, 4.64601145253590350367853365568, 6.51538029548541644174973433940, 7.35768218992449691366728637217, 8.009542888860805966129963709488, 9.587977919539543459127680942726, 10.20880373963852354335606416469, 11.53268217120850481581690866508, 12.30144446414001116480138124870

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