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Properties

Label 2-192-12.11-c3-0-19
Degree $2$
Conductor $192$
Sign $-0.666 + 0.745i$
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  + (3.46 − 3.87i)3-s − 8.94i·5-s − 7.74i·7-s + (−3.00 − 26.8i)9-s − 34.6·11-s + 10·13-s + (−34.6 − 30.9i)15-s + 35.7i·17-s − 69.7i·19-s + (−30.0 − 26.8i)21-s − 96.9·23-s + 44.9·25-s + (−114. − 81.3i)27-s + 152. i·29-s − 224. i·31-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)3-s − 0.799i·5-s − 0.418i·7-s + (−0.111 − 0.993i)9-s − 0.949·11-s + 0.213·13-s + (−0.596 − 0.533i)15-s + 0.510i·17-s − 0.841i·19-s + (−0.311 − 0.278i)21-s − 0.879·23-s + 0.359·25-s + (−0.814 − 0.579i)27-s + 0.973i·29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.666 + 0.745i$
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: \((2,\ 192,\ (\ :3/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.707576 - 1.58218i\)
\(L(\frac12)\) \(\approx\) \(0.707576 - 1.58218i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.46 + 3.87i)T \)
good5 \( 1 + 8.94iT - 125T^{2} \)
7 \( 1 + 7.74iT - 343T^{2} \)
11 \( 1 + 34.6T + 1.33e3T^{2} \)
13 \( 1 - 10T + 2.19e3T^{2} \)
17 \( 1 - 35.7iT - 4.91e3T^{2} \)
19 \( 1 + 69.7iT - 6.85e3T^{2} \)
23 \( 1 + 96.9T + 1.21e4T^{2} \)
29 \( 1 - 152. iT - 2.43e4T^{2} \)
31 \( 1 + 224. iT - 2.97e4T^{2} \)
37 \( 1 - 130T + 5.06e4T^{2} \)
41 \( 1 + 125. iT - 6.89e4T^{2} \)
43 \( 1 + 224. iT - 7.95e4T^{2} \)
47 \( 1 + 193.T + 1.03e5T^{2} \)
53 \( 1 + 545. iT - 1.48e5T^{2} \)
59 \( 1 + 173.T + 2.05e5T^{2} \)
61 \( 1 - 442T + 2.26e5T^{2} \)
67 \( 1 - 735. iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 - 410T + 3.89e5T^{2} \)
79 \( 1 - 85.2iT - 4.93e5T^{2} \)
83 \( 1 - 1.25e3T + 5.71e5T^{2} \)
89 \( 1 - 840. iT - 7.04e5T^{2} \)
97 \( 1 - 770T + 9.12e5T^{2} \)
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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

−11.97352220073801066930436332015, −10.74121061350178507540548305547, −9.532521520739510069986357155773, −8.528029062007044203068344692902, −7.79351034233387576704676272556, −6.66450691492483079038077022957, −5.28900633405240212186874735122, −3.83823001705736084187568158772, −2.26707354860444140372591125598, −0.69012137169029699312114267735, 2.36793381327493036202882384909, 3.39222402185979479726926074389, 4.82179479239538230924550816847, 6.08329952447792645773807020609, 7.57014018808288697012341934904, 8.395183939217363798842259757106, 9.608059194473581816010925607919, 10.38446943324474369155789705924, 11.20094408096995853542793918940, 12.45495795642981486611832218136

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