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Properties

Label 2-192-16.13-c3-0-9
Degree $2$
Conductor $192$
Sign $0.548 + 0.836i$
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 + (11.7 − 11.7i)5-s − 12.5i·7-s + 8.99i·9-s + (17.0 − 17.0i)11-s + (−49.2 − 49.2i)13-s + 50.0·15-s − 51.8·17-s + (11.6 + 11.6i)19-s + (26.6 − 26.6i)21-s − 74.5i·23-s − 153. i·25-s + (−19.0 + 19.0i)27-s + (211. + 211. i)29-s + 326.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.05 − 1.05i)5-s − 0.679i·7-s + 0.333i·9-s + (0.466 − 0.466i)11-s + (−1.05 − 1.05i)13-s + 0.861·15-s − 0.739·17-s + (0.140 + 0.140i)19-s + (0.277 − 0.277i)21-s − 0.675i·23-s − 1.22i·25-s + (−0.136 + 0.136i)27-s + (1.35 + 1.35i)29-s + 1.88·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.548 + 0.836i$
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: \((2,\ 192,\ (\ :3/2),\ 0.548 + 0.836i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.95374 - 1.05514i\)
\(L(\frac12)\) \(\approx\) \(1.95374 - 1.05514i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.7 + 11.7i)T - 125iT^{2} \)
7 \( 1 + 12.5iT - 343T^{2} \)
11 \( 1 + (-17.0 + 17.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (49.2 + 49.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 51.8T + 4.91e3T^{2} \)
19 \( 1 + (-11.6 - 11.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 74.5iT - 1.21e4T^{2} \)
29 \( 1 + (-211. - 211. i)T + 2.43e4iT^{2} \)
31 \( 1 - 326.T + 2.97e4T^{2} \)
37 \( 1 + (-110. + 110. i)T - 5.06e4iT^{2} \)
41 \( 1 + 348. iT - 6.89e4T^{2} \)
43 \( 1 + (205. - 205. i)T - 7.95e4iT^{2} \)
47 \( 1 + 254.T + 1.03e5T^{2} \)
53 \( 1 + (225. - 225. i)T - 1.48e5iT^{2} \)
59 \( 1 + (285. - 285. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-286. - 286. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-627. - 627. i)T + 3.00e5iT^{2} \)
71 \( 1 - 274. iT - 3.57e5T^{2} \)
73 \( 1 + 298. iT - 3.89e5T^{2} \)
79 \( 1 - 175.T + 4.93e5T^{2} \)
83 \( 1 + (125. + 125. i)T + 5.71e5iT^{2} \)
89 \( 1 - 900. iT - 7.04e5T^{2} \)
97 \( 1 - 5.27T + 9.12e5T^{2} \)
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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

−12.13550840411295932494645252255, −10.59231896566191947011361705918, −9.943368751380713204552154003515, −8.968444996374917979992151472858, −8.136838306980607099274879020892, −6.65514723651836978241727462845, −5.31746384633685278186098184505, −4.41955137338626121544336522831, −2.70999735016914728561708016160, −0.974896262582260585158975970089, 1.96586802836019182818933582634, 2.78614834408316577496201886430, 4.69605970852984043579943392620, 6.35489469005762221691143333513, 6.76619762644681858108798032270, 8.184984798370553220561693206902, 9.548249635960234583438526607039, 9.886577766138382758770204366235, 11.43256324539062900601732522113, 12.14099348003019710502152479304

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