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Properties

Label 2-192-48.5-c2-0-8
Degree $2$
Conductor $192$
Sign $0.998 - 0.0557i$
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  + (2.99 − 0.164i)3-s + (3.61 − 3.61i)5-s + 12.2i·7-s + (8.94 − 0.985i)9-s + (1.76 − 1.76i)11-s + (−2.38 + 2.38i)13-s + (10.2 − 11.4i)15-s − 20.0i·17-s + (8.77 − 8.77i)19-s + (2.02 + 36.7i)21-s − 13.1·23-s − 1.10i·25-s + (26.6 − 4.42i)27-s + (6.51 + 6.51i)29-s − 37.5·31-s + ⋯
L(s)  = 1  + (0.998 − 0.0548i)3-s + (0.722 − 0.722i)5-s + 1.75i·7-s + (0.993 − 0.109i)9-s + (0.160 − 0.160i)11-s + (−0.183 + 0.183i)13-s + (0.681 − 0.761i)15-s − 1.18i·17-s + (0.461 − 0.461i)19-s + (0.0962 + 1.75i)21-s − 0.573·23-s − 0.0443i·25-s + (0.986 − 0.163i)27-s + (0.224 + 0.224i)29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.998 - 0.0557i$
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: \((2,\ 192,\ (\ :1),\ 0.998 - 0.0557i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31399 + 0.0645655i\)
\(L(\frac12)\) \(\approx\) \(2.31399 + 0.0645655i\)
\(L(2)\) 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 \( 1 + (-2.99 + 0.164i)T \)
good5 \( 1 + (-3.61 + 3.61i)T - 25iT^{2} \)
7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + (-1.76 + 1.76i)T - 121iT^{2} \)
13 \( 1 + (2.38 - 2.38i)T - 169iT^{2} \)
17 \( 1 + 20.0iT - 289T^{2} \)
19 \( 1 + (-8.77 + 8.77i)T - 361iT^{2} \)
23 \( 1 + 13.1T + 529T^{2} \)
29 \( 1 + (-6.51 - 6.51i)T + 841iT^{2} \)
31 \( 1 + 37.5T + 961T^{2} \)
37 \( 1 + (-10.0 - 10.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 4.57T + 1.68e3T^{2} \)
43 \( 1 + (21.2 + 21.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (21.5 - 21.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (53.6 - 53.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (19.2 - 19.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (31.5 - 31.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 65.1T + 5.04e3T^{2} \)
73 \( 1 + 50.2iT - 5.32e3T^{2} \)
79 \( 1 + 20.9T + 6.24e3T^{2} \)
83 \( 1 + (6.35 + 6.35i)T + 6.88e3iT^{2} \)
89 \( 1 + 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−12.42764844648403780583368189982, −11.61840306431347470198029900136, −9.883286463474540398983896848765, −9.083596467774821762160468115382, −8.722730520213271584990545943306, −7.32182334251882721384060716649, −5.84391781734699650364518443267, −4.86223868135095045315187454403, −2.98309300287836959429584555936, −1.85531464722611571256412654944, 1.68254723178587508416361093605, 3.32838672582373870521564437517, 4.33079657256297638839602495358, 6.25524276082819315838745822264, 7.29784732886475053346595585233, 8.067599292277973001420931785266, 9.573073372306071473578765475488, 10.22704864843151422760157810843, 10.90832147910241433310129003677, 12.65388391644627078303575089772

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