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Properties

Label 2-192-48.11-c3-0-0
Degree $2$
Conductor $192$
Sign $-0.566 - 0.824i$
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  + (0.151 − 5.19i)3-s + (−4.66 + 4.66i)5-s − 0.405·7-s + (−26.9 − 1.56i)9-s + (−5.82 − 5.82i)11-s + (−35.2 + 35.2i)13-s + (23.5 + 24.9i)15-s + 49.3i·17-s + (−108. − 108. i)19-s + (−0.0612 + 2.10i)21-s + 130. i·23-s + 81.4i·25-s + (−12.2 + 139. i)27-s + (172. + 172. i)29-s + 36.1i·31-s + ⋯
L(s)  = 1  + (0.0290 − 0.999i)3-s + (−0.417 + 0.417i)5-s − 0.0219·7-s + (−0.998 − 0.0581i)9-s + (−0.159 − 0.159i)11-s + (−0.751 + 0.751i)13-s + (0.405 + 0.429i)15-s + 0.703i·17-s + (−1.30 − 1.30i)19-s + (−0.000636 + 0.0219i)21-s + 1.18i·23-s + 0.651i·25-s + (−0.0870 + 0.996i)27-s + (1.10 + 1.10i)29-s + 0.209i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.566 - 0.824i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.566 - 0.824i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.127405 + 0.242147i\)
\(L(\frac12)\) \(\approx\) \(0.127405 + 0.242147i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.151 + 5.19i)T \)
good5 \( 1 + (4.66 - 4.66i)T - 125iT^{2} \)
7 \( 1 + 0.405T + 343T^{2} \)
11 \( 1 + (5.82 + 5.82i)T + 1.33e3iT^{2} \)
13 \( 1 + (35.2 - 35.2i)T - 2.19e3iT^{2} \)
17 \( 1 - 49.3iT - 4.91e3T^{2} \)
19 \( 1 + (108. + 108. i)T + 6.85e3iT^{2} \)
23 \( 1 - 130. iT - 1.21e4T^{2} \)
29 \( 1 + (-172. - 172. i)T + 2.43e4iT^{2} \)
31 \( 1 - 36.1iT - 2.97e4T^{2} \)
37 \( 1 + (257. + 257. i)T + 5.06e4iT^{2} \)
41 \( 1 - 5.87T + 6.89e4T^{2} \)
43 \( 1 + (170. - 170. i)T - 7.95e4iT^{2} \)
47 \( 1 + 181.T + 1.03e5T^{2} \)
53 \( 1 + (-148. + 148. i)T - 1.48e5iT^{2} \)
59 \( 1 + (567. + 567. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-481. + 481. i)T - 2.26e5iT^{2} \)
67 \( 1 + (296. + 296. i)T + 3.00e5iT^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 + 528. iT - 4.93e5T^{2} \)
83 \( 1 + (-713. + 713. i)T - 5.71e5iT^{2} \)
89 \( 1 + 204.T + 7.04e5T^{2} \)
97 \( 1 - 275.T + 9.12e5T^{2} \)
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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

−12.48134601419319400743332738323, −11.48412548123484101059818935978, −10.75674817513671492033755977930, −9.240478260932563720635516163212, −8.260849374817806631896914873127, −7.15785223435838000375910963524, −6.50848876731306498514290560694, −4.99990583401243905743763859017, −3.30029061826482581824053002972, −1.87331132917904278721191185409, 0.11261427026631422184189130635, 2.68325534151667758651300743554, 4.17109053647265896864818623086, 5.01209502185114570307343069438, 6.33938070387149555099781889003, 7.997889885897288960013985486538, 8.642566608682191993152595796281, 10.03691801210304036980762665223, 10.42974164863599651797763487611, 11.84108943306855418453293674573

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