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Properties

Label 2-192-16.13-c3-0-5
Degree $2$
Conductor $192$
Sign $0.334 - 0.942i$
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 + (2.24 − 2.24i)5-s + 9.00i·7-s + 8.99i·9-s + (−11.0 + 11.0i)11-s + (54.5 + 54.5i)13-s + 9.51·15-s + 44.0·17-s + (−49.9 − 49.9i)19-s + (−19.0 + 19.0i)21-s + 117. i·23-s + 114. i·25-s + (−19.0 + 19.0i)27-s + (40.6 + 40.6i)29-s + 196.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.200 − 0.200i)5-s + 0.486i·7-s + 0.333i·9-s + (−0.302 + 0.302i)11-s + (1.16 + 1.16i)13-s + 0.163·15-s + 0.627·17-s + (−0.603 − 0.603i)19-s + (−0.198 + 0.198i)21-s + 1.06i·23-s + 0.919i·25-s + (−0.136 + 0.136i)27-s + (0.260 + 0.260i)29-s + 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.334 - 0.942i$
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: \((2,\ 192,\ (\ :3/2),\ 0.334 - 0.942i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.62244 + 1.14511i\)
\(L(\frac12)\) \(\approx\) \(1.62244 + 1.14511i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.24 + 2.24i)T - 125iT^{2} \)
7 \( 1 - 9.00iT - 343T^{2} \)
11 \( 1 + (11.0 - 11.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-54.5 - 54.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 44.0T + 4.91e3T^{2} \)
19 \( 1 + (49.9 + 49.9i)T + 6.85e3iT^{2} \)
23 \( 1 - 117. iT - 1.21e4T^{2} \)
29 \( 1 + (-40.6 - 40.6i)T + 2.43e4iT^{2} \)
31 \( 1 - 196.T + 2.97e4T^{2} \)
37 \( 1 + (248. - 248. i)T - 5.06e4iT^{2} \)
41 \( 1 + 457. iT - 6.89e4T^{2} \)
43 \( 1 + (204. - 204. i)T - 7.95e4iT^{2} \)
47 \( 1 - 390.T + 1.03e5T^{2} \)
53 \( 1 + (138. - 138. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-263. + 263. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-29.1 - 29.1i)T + 2.26e5iT^{2} \)
67 \( 1 + (508. + 508. i)T + 3.00e5iT^{2} \)
71 \( 1 + 788. iT - 3.57e5T^{2} \)
73 \( 1 - 92.2iT - 3.89e5T^{2} \)
79 \( 1 - 174.T + 4.93e5T^{2} \)
83 \( 1 + (914. + 914. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.45e3iT - 7.04e5T^{2} \)
97 \( 1 + 229.T + 9.12e5T^{2} \)
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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

−12.22355436230043182642401126307, −11.28591238748854916392271196278, −10.21340029972608163891499794005, −9.150064012182955299808867206746, −8.535707950376763213378301172978, −7.16383193496465131952483156487, −5.87183197810363421619371165921, −4.66296660618799271431945737146, −3.33696718546601832524114742162, −1.73484924730475619026943769355, 0.898302229377552123036987904292, 2.70040845549179979517948542944, 3.97539310408162815477992980149, 5.69594591136169194130556146564, 6.67056903190494833217639948008, 8.009136600753974761777581706059, 8.557924802935557218275082790039, 10.17075557491208626006525256771, 10.65091511812035961062668840622, 12.05178848130307398616125674862

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