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Properties

Label 2-192-48.35-c3-0-0
Degree $2$
Conductor $192$
Sign $-0.350 - 0.936i$
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  + (0.0749 − 5.19i)3-s + (−5.37 − 5.37i)5-s − 14.8·7-s + (−26.9 − 0.779i)9-s + (−30.0 + 30.0i)11-s + (61.5 + 61.5i)13-s + (−28.3 + 27.5i)15-s − 48.8i·17-s + (−7.45 + 7.45i)19-s + (−1.11 + 77.1i)21-s + 43.0i·23-s − 67.1i·25-s + (−6.07 + 140. i)27-s + (−32.9 + 32.9i)29-s + 173. i·31-s + ⋯
L(s)  = 1  + (0.0144 − 0.999i)3-s + (−0.480 − 0.480i)5-s − 0.802·7-s + (−0.999 − 0.0288i)9-s + (−0.823 + 0.823i)11-s + (1.31 + 1.31i)13-s + (−0.487 + 0.473i)15-s − 0.696i·17-s + (−0.0900 + 0.0900i)19-s + (−0.0115 + 0.802i)21-s + 0.390i·23-s − 0.537i·25-s + (−0.0432 + 0.999i)27-s + (−0.211 + 0.211i)29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.120098 + 0.173219i\)
\(L(\frac12)\) \(\approx\) \(0.120098 + 0.173219i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.0749 + 5.19i)T \)
good5 \( 1 + (5.37 + 5.37i)T + 125iT^{2} \)
7 \( 1 + 14.8T + 343T^{2} \)
11 \( 1 + (30.0 - 30.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-61.5 - 61.5i)T + 2.19e3iT^{2} \)
17 \( 1 + 48.8iT - 4.91e3T^{2} \)
19 \( 1 + (7.45 - 7.45i)T - 6.85e3iT^{2} \)
23 \( 1 - 43.0iT - 1.21e4T^{2} \)
29 \( 1 + (32.9 - 32.9i)T - 2.43e4iT^{2} \)
31 \( 1 - 173. iT - 2.97e4T^{2} \)
37 \( 1 + (177. - 177. i)T - 5.06e4iT^{2} \)
41 \( 1 + 454.T + 6.89e4T^{2} \)
43 \( 1 + (239. + 239. i)T + 7.95e4iT^{2} \)
47 \( 1 + 30.4T + 1.03e5T^{2} \)
53 \( 1 + (-235. - 235. i)T + 1.48e5iT^{2} \)
59 \( 1 + (260. - 260. i)T - 2.05e5iT^{2} \)
61 \( 1 + (388. + 388. i)T + 2.26e5iT^{2} \)
67 \( 1 + (334. - 334. i)T - 3.00e5iT^{2} \)
71 \( 1 + 522. iT - 3.57e5T^{2} \)
73 \( 1 + 689. iT - 3.89e5T^{2} \)
79 \( 1 + 692. iT - 4.93e5T^{2} \)
83 \( 1 + (-677. - 677. i)T + 5.71e5iT^{2} \)
89 \( 1 - 261.T + 7.04e5T^{2} \)
97 \( 1 + 641.T + 9.12e5T^{2} \)
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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

−12.28944362922285765985864206143, −11.78697150932590729329536526021, −10.49797428106309188856438011762, −9.154061270142834531909435019499, −8.329910356689062415626012460914, −7.15268440390424408036089502616, −6.38390587401716926846426688639, −4.92412417560349813880165307851, −3.34021752928933806963021807893, −1.67909111966995395150754456358, 0.089985197059597934133344280965, 3.08645685403308174974617907759, 3.70480640461604313135418352436, 5.41284114458614990382055656290, 6.29806868990163245079235830357, 7.958272901086811850949402554723, 8.714888378367830831474390204834, 10.05753266386794950158774735001, 10.71497476968494556762505863442, 11.42264366930159713970073654482

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