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Properties

Label 2-192-16.11-c2-0-0
Degree $2$
Conductor $192$
Sign $-0.997 - 0.0662i$
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  + (1.22 + 1.22i)3-s + (−3.40 − 3.40i)5-s − 12.1·7-s + 2.99i·9-s + (−9.81 + 9.81i)11-s + (−7.76 + 7.76i)13-s − 8.34i·15-s + 9.73·17-s + (−11.2 − 11.2i)19-s + (−14.8 − 14.8i)21-s + 20.2·23-s − 1.80i·25-s + (−3.67 + 3.67i)27-s + (−16.4 + 16.4i)29-s − 26.3i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.681 − 0.681i)5-s − 1.73·7-s + 0.333i·9-s + (−0.891 + 0.891i)11-s + (−0.597 + 0.597i)13-s − 0.556i·15-s + 0.572·17-s + (−0.593 − 0.593i)19-s + (−0.707 − 0.707i)21-s + 0.881·23-s − 0.0720i·25-s + (−0.136 + 0.136i)27-s + (−0.565 + 0.565i)29-s − 0.850i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0662i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.997 - 0.0662i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.997 - 0.0662i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00504284 + 0.151963i\)
\(L(\frac12)\) \(\approx\) \(0.00504284 + 0.151963i\)
\(L(2)\) 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 \( 1 + (3.40 + 3.40i)T + 25iT^{2} \)
7 \( 1 + 12.1T + 49T^{2} \)
11 \( 1 + (9.81 - 9.81i)T - 121iT^{2} \)
13 \( 1 + (7.76 - 7.76i)T - 169iT^{2} \)
17 \( 1 - 9.73T + 289T^{2} \)
19 \( 1 + (11.2 + 11.2i)T + 361iT^{2} \)
23 \( 1 - 20.2T + 529T^{2} \)
29 \( 1 + (16.4 - 16.4i)T - 841iT^{2} \)
31 \( 1 + 26.3iT - 961T^{2} \)
37 \( 1 + (23.7 + 23.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 24.7iT - 1.68e3T^{2} \)
43 \( 1 + (29.8 - 29.8i)T - 1.84e3iT^{2} \)
47 \( 1 + 31.3iT - 2.20e3T^{2} \)
53 \( 1 + (-36.8 - 36.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-14.1 + 14.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (42.5 - 42.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (48.7 + 48.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 7.73T + 5.04e3T^{2} \)
73 \( 1 + 85.4iT - 5.32e3T^{2} \)
79 \( 1 - 105. iT - 6.24e3T^{2} \)
83 \( 1 + (-62.1 - 62.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−12.77460118110998684663027437612, −12.08738648932302011860388905217, −10.61276067270233513902236208590, −9.668337552960832574472716882714, −9.015880790437236542211255659024, −7.71754418294938102197915580992, −6.73614175783776483254368323659, −5.12682412774171062279949437632, −4.00000017729265828664809107135, −2.69454776926109546435675362522, 0.07820144796162669404181266296, 2.90546082461030440769227356108, 3.49773083244234130832672353848, 5.62735887836375337897775747459, 6.77622655999799823821361621761, 7.61166567446337804701105862602, 8.709778604604584086362884989307, 9.965705735808568914524334964065, 10.69560057172906480023494282276, 12.02536010270812687167929552603

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