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Properties

Label 2-192-24.11-c1-0-6
Degree $2$
Conductor $192$
Sign $0.965 + 0.258i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s − 4i·7-s + 2.99·9-s + 6.92i·13-s − 3.46·19-s − 6.92i·21-s − 5·25-s + 5.19·27-s + 4i·31-s − 6.92i·37-s + 11.9i·39-s − 10.3·43-s − 9·49-s − 5.99·57-s + 6.92i·61-s + ⋯
L(s)  = 1  + 1.00·3-s − 1.51i·7-s + 0.999·9-s + 1.92i·13-s − 0.794·19-s − 1.51i·21-s − 25-s + 1.00·27-s + 0.718i·31-s − 1.13i·37-s + 1.92i·39-s − 1.58·43-s − 1.28·49-s − 0.794·57-s + 0.887i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55724 - 0.205014i\)
\(L(\frac12)\) \(\approx\) \(1.55724 - 0.205014i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−12.72688765793722944138524159062, −11.44449129393493554793062099954, −10.37336399409895430844143666559, −9.504743568209376636537509326563, −8.493713848901887557116549323899, −7.34484756855076761291295593840, −6.64656850861725757157032661674, −4.47368175568330826944446747732, −3.73695545407449021410931164443, −1.84180163242471539062006225125, 2.26206078522265311928677484590, 3.37496643694661587025125649963, 5.10405922336065834619985805852, 6.26500701521122526358390210473, 7.898146322273252359552565351482, 8.431805541985900176287300325960, 9.500973331588086930755416856244, 10.39158398187714793047870331352, 11.79418472679707024965317167556, 12.74886822754527569849731786917

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