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Properties

Label 2-192-3.2-c2-0-0
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

\(L(\frac{3}{2})\) \(\approx\) \(0.0440522 - 0.175863i\)
\(L(\frac12)\) \(\approx\) \(0.0440522 - 0.175863i\)
\(L(2)\) 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

−12.75036662506769114573545260698, −11.61041272090993498315965704444, −10.85825559791518553308835389255, −10.21699412196717432550178196443, −9.181859514434677727198227347599, −7.49458689386566053176912035398, −6.42965121271951234050653037521, −5.81969621941266909414881572127, −4.03256220575157467098998901306, −2.86409261275417217904979543644, 0.11136600134054828665966240242, 1.88794707335394461039887543644, 4.40894961992028510431517457752, 5.17294950887508157696896726092, 6.47399744890267992836552336299, 7.48087529083086005919033918143, 8.783553357866111495385807092185, 9.762106987276690578847406369218, 10.76886171697768760479996516101, 12.25749887958143053903097787535

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