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Properties

Label 2-192-16.13-c3-0-8
Degree $2$
Conductor $192$
Sign $0.999 + 0.00522i$
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  + (2.12 + 2.12i)3-s + (−3.72 + 3.72i)5-s − 20.2i·7-s + 8.99i·9-s + (47.5 − 47.5i)11-s + (27.8 + 27.8i)13-s − 15.8·15-s + 56.3·17-s + (66.1 + 66.1i)19-s + (42.9 − 42.9i)21-s + 3.66i·23-s + 97.2i·25-s + (−19.0 + 19.0i)27-s + (−86.8 − 86.8i)29-s + 102.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.333 + 0.333i)5-s − 1.09i·7-s + 0.333i·9-s + (1.30 − 1.30i)11-s + (0.594 + 0.594i)13-s − 0.271·15-s + 0.804·17-s + (0.798 + 0.798i)19-s + (0.446 − 0.446i)21-s + 0.0332i·23-s + 0.778i·25-s + (−0.136 + 0.136i)27-s + (−0.556 − 0.556i)29-s + 0.595·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00522i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.999 + 0.00522i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.999 + 0.00522i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.07059 - 0.00540935i\)
\(L(\frac12)\) \(\approx\) \(2.07059 - 0.00540935i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.12 - 2.12i)T \)
good5 \( 1 + (3.72 - 3.72i)T - 125iT^{2} \)
7 \( 1 + 20.2iT - 343T^{2} \)
11 \( 1 + (-47.5 + 47.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (-27.8 - 27.8i)T + 2.19e3iT^{2} \)
17 \( 1 - 56.3T + 4.91e3T^{2} \)
19 \( 1 + (-66.1 - 66.1i)T + 6.85e3iT^{2} \)
23 \( 1 - 3.66iT - 1.21e4T^{2} \)
29 \( 1 + (86.8 + 86.8i)T + 2.43e4iT^{2} \)
31 \( 1 - 102.T + 2.97e4T^{2} \)
37 \( 1 + (-66.8 + 66.8i)T - 5.06e4iT^{2} \)
41 \( 1 - 29.5iT - 6.89e4T^{2} \)
43 \( 1 + (-372. + 372. i)T - 7.95e4iT^{2} \)
47 \( 1 + 539.T + 1.03e5T^{2} \)
53 \( 1 + (-385. + 385. i)T - 1.48e5iT^{2} \)
59 \( 1 + (71.0 - 71.0i)T - 2.05e5iT^{2} \)
61 \( 1 + (155. + 155. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-178. - 178. i)T + 3.00e5iT^{2} \)
71 \( 1 + 483. iT - 3.57e5T^{2} \)
73 \( 1 - 908. iT - 3.89e5T^{2} \)
79 \( 1 + 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + (871. + 871. i)T + 5.71e5iT^{2} \)
89 \( 1 - 185. iT - 7.04e5T^{2} \)
97 \( 1 + 725.T + 9.12e5T^{2} \)
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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

−11.76862455858998048110140122608, −11.15994282498859857197431672403, −10.09989100079577169602461954939, −9.117866504431170445384119987476, −8.040835205220455701621640712807, −7.01018721864073410295394242687, −5.78359688004750776218414225823, −3.97206449880114786160544940808, −3.47034941707068198825256000159, −1.14304529496338804967199327362, 1.32536922354208885960371079923, 2.90142521341551181945566838674, 4.40641856908184195640520620928, 5.79717745979419947719383885455, 6.99414244301760158659321036600, 8.096608608970121395834594753087, 9.053081095126697894534880945808, 9.804589312282684181105425109966, 11.43222564292506116622394746363, 12.21265109595151179334292585035

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