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Properties

Label 2-192-16.5-c3-0-3
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: $\chi_{192} (49, \cdot )$
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

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

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