Properties

Degree $2$
Conductor $192$
Sign $-0.949 - 0.313i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (−5.24 − 5.24i)5-s + 5.32·7-s + 2.99i·9-s + (−12.2 + 12.2i)11-s + (−5.73 + 5.73i)13-s + 12.8i·15-s − 23.3·17-s + (−11.7 − 11.7i)19-s + (−6.52 − 6.52i)21-s − 5.80·23-s + 29.9i·25-s + (3.67 − 3.67i)27-s + (18.3 − 18.3i)29-s − 16.9i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.04 − 1.04i)5-s + 0.761·7-s + 0.333i·9-s + (−1.11 + 1.11i)11-s + (−0.441 + 0.441i)13-s + 0.856i·15-s − 1.37·17-s + (−0.618 − 0.618i)19-s + (−0.310 − 0.310i)21-s − 0.252·23-s + 1.19i·25-s + (0.136 − 0.136i)27-s + (0.634 − 0.634i)29-s − 0.545i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.949 - 0.313i$
Motivic weight: \(2\)
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.949 - 0.313i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0266466 + 0.165472i\)
\(L(\frac12)\) \(\approx\) \(0.0266466 + 0.165472i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (5.24 + 5.24i)T + 25iT^{2} \)
7 \( 1 - 5.32T + 49T^{2} \)
11 \( 1 + (12.2 - 12.2i)T - 121iT^{2} \)
13 \( 1 + (5.73 - 5.73i)T - 169iT^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 + (11.7 + 11.7i)T + 361iT^{2} \)
23 \( 1 + 5.80T + 529T^{2} \)
29 \( 1 + (-18.3 + 18.3i)T - 841iT^{2} \)
31 \( 1 + 16.9iT - 961T^{2} \)
37 \( 1 + (-15.3 - 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (33.4 - 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (66.9 + 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-27.1 + 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-65.2 + 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-37.6 - 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (24.1 + 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−11.74027691938237156499638304853, −11.15283521738275615511316275547, −9.773750196316962608678311859974, −8.412755806251705721125578348949, −7.82851694649397731240304040279, −6.68512021366577302865681893883, −4.82764789073500026513148400685, −4.56839909363678363806763179654, −2.08568911128264950704459460023, −0.096982338384529576042743917824, 2.79426399899081875995885310821, 4.10282554883283270653495781771, 5.35983012177411974922426933905, 6.69112011725867990265899081092, 7.82770991646654755852297979891, 8.612222564947000350840244507015, 10.39637894081620131989980069653, 10.87553859843686604869183638235, 11.54622531709065900245352514038, 12.67243580695779532700893121228

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