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Properties

Label 2-192-48.5-c2-0-5
Degree $2$
Conductor $192$
Sign $-0.321 - 0.947i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 2.75i)3-s + (−0.00985 + 0.00985i)5-s + 6.42i·7-s + (−6.19 + 6.53i)9-s + (−9.07 + 9.07i)11-s + (12.6 − 12.6i)13-s + (−0.0388 − 0.0154i)15-s + 19.0i·17-s + (2.07 − 2.07i)19-s + (−17.7 + 7.61i)21-s − 19.5·23-s + 24.9i·25-s + (−25.3 − 9.32i)27-s + (−11.1 − 11.1i)29-s + 59.9·31-s + ⋯
L(s)  = 1  + (0.395 + 0.918i)3-s + (−0.00197 + 0.00197i)5-s + 0.917i·7-s + (−0.687 + 0.725i)9-s + (−0.824 + 0.824i)11-s + (0.969 − 0.969i)13-s + (−0.00259 − 0.00103i)15-s + 1.11i·17-s + (0.109 − 0.109i)19-s + (−0.842 + 0.362i)21-s − 0.850·23-s + 0.999i·25-s + (−0.938 − 0.345i)27-s + (−0.385 − 0.385i)29-s + 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.321 - 0.947i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.321 - 0.947i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.881837 + 1.23025i\)
\(L(\frac12)\) \(\approx\) \(0.881837 + 1.23025i\)
\(L(2)\) 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 \( 1 + (-1.18 - 2.75i)T \)
good5 \( 1 + (0.00985 - 0.00985i)T - 25iT^{2} \)
7 \( 1 - 6.42iT - 49T^{2} \)
11 \( 1 + (9.07 - 9.07i)T - 121iT^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 - 19.0iT - 289T^{2} \)
19 \( 1 + (-2.07 + 2.07i)T - 361iT^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 + (11.1 + 11.1i)T + 841iT^{2} \)
31 \( 1 - 59.9T + 961T^{2} \)
37 \( 1 + (-9.32 - 9.32i)T + 1.36e3iT^{2} \)
41 \( 1 - 47.2T + 1.68e3T^{2} \)
43 \( 1 + (24.1 + 24.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (-20.6 + 20.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (-60.3 + 60.3i)T - 3.48e3iT^{2} \)
61 \( 1 + (-48.0 + 48.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-23.7 + 23.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 13.5T + 5.04e3T^{2} \)
73 \( 1 + 31.4iT - 5.32e3T^{2} \)
79 \( 1 + 47.4T + 6.24e3T^{2} \)
83 \( 1 + (-70.3 - 70.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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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

−12.68810949154885600152578647128, −11.48792300968316337365890779652, −10.45566930271031897620783349201, −9.749600627684218257388355946032, −8.543216983908680269258117098042, −7.915156586726668093966019740904, −6.03823665236070955710686150596, −5.11445244029067646770925010435, −3.72690148092260040582632617605, −2.38120190730707950753540799747, 0.871309899927892584647527645303, 2.69374369629990885550647017309, 4.14414182418841394457077365008, 5.90851290576879676195185613410, 6.90780300551986091600965036615, 7.917457634524312396212809642527, 8.753286955748642366610546107878, 10.04405609718619486782553107157, 11.20727416950072416485968119866, 11.98738638937295778110199269460

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