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Properties

Label 2-192-12.11-c3-0-14
Degree $2$
Conductor $192$
Sign $0.775 + 0.631i$
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  + (3.28 − 4.02i)3-s + 2.00i·5-s + 14.5i·7-s + (−5.45 − 26.4i)9-s + 50.3·11-s + 71.3·13-s + (8.05 + 6.56i)15-s − 64.1i·17-s − 21.1i·19-s + (58.4 + 47.6i)21-s − 129.·23-s + 120.·25-s + (−124. − 64.8i)27-s − 295. i·29-s + 71.7i·31-s + ⋯
L(s)  = 1  + (0.631 − 0.775i)3-s + 0.178i·5-s + 0.783i·7-s + (−0.201 − 0.979i)9-s + 1.37·11-s + 1.52·13-s + (0.138 + 0.113i)15-s − 0.915i·17-s − 0.255i·19-s + (0.607 + 0.494i)21-s − 1.17·23-s + 0.967·25-s + (−0.886 − 0.462i)27-s − 1.89i·29-s + 0.415i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.775 + 0.631i$
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: \((2,\ 192,\ (\ :3/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.20592 - 0.784910i\)
\(L(\frac12)\) \(\approx\) \(2.20592 - 0.784910i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.28 + 4.02i)T \)
good5 \( 1 - 2.00iT - 125T^{2} \)
7 \( 1 - 14.5iT - 343T^{2} \)
11 \( 1 - 50.3T + 1.33e3T^{2} \)
13 \( 1 - 71.3T + 2.19e3T^{2} \)
17 \( 1 + 64.1iT - 4.91e3T^{2} \)
19 \( 1 + 21.1iT - 6.85e3T^{2} \)
23 \( 1 + 129.T + 1.21e4T^{2} \)
29 \( 1 + 295. iT - 2.43e4T^{2} \)
31 \( 1 - 71.7iT - 2.97e4T^{2} \)
37 \( 1 - 33.9T + 5.06e4T^{2} \)
41 \( 1 - 315. iT - 6.89e4T^{2} \)
43 \( 1 - 474. iT - 7.95e4T^{2} \)
47 \( 1 - 116.T + 1.03e5T^{2} \)
53 \( 1 + 219. iT - 1.48e5T^{2} \)
59 \( 1 - 495.T + 2.05e5T^{2} \)
61 \( 1 + 224.T + 2.26e5T^{2} \)
67 \( 1 + 127. iT - 3.00e5T^{2} \)
71 \( 1 + 526.T + 3.57e5T^{2} \)
73 \( 1 + 526.T + 3.89e5T^{2} \)
79 \( 1 - 1.28e3iT - 4.93e5T^{2} \)
83 \( 1 + 357.T + 5.71e5T^{2} \)
89 \( 1 + 328. iT - 7.04e5T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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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

−11.88727129102287589338293478582, −11.39856532715040607320157658229, −9.677874021776168351213204567131, −8.867373233381201222024692764260, −8.045662549701458222857389541947, −6.67100615871894899589259800766, −6.00531164494291083577169634869, −4.03764708642250432854784493440, −2.71495871858190289210028793477, −1.21672723353481545371116600363, 1.48051869534338500175813300863, 3.61005827124136163723386492477, 4.15326203312710367345832369052, 5.80537798755938489421931948821, 7.09600224478478507107651668979, 8.506699239788362146319999893534, 8.999922303835848559400867735307, 10.34413184301566335313711136773, 10.88143892658884258783058714703, 12.17173494711337475772018691644

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