Properties

Label 2-192-3.2-c2-0-5
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·7-s + 9·9-s + 22·13-s + 26·19-s + 6·21-s + 25·25-s − 27·27-s + 46·31-s − 26·37-s − 66·39-s − 22·43-s − 45·49-s − 78·57-s − 74·61-s − 18·63-s + 122·67-s − 46·73-s − 75·75-s + 142·79-s + 81·81-s − 44·91-s − 138·93-s + 2·97-s − 194·103-s + 214·109-s + 78·111-s + ⋯
L(s)  = 1  − 3-s − 2/7·7-s + 9-s + 1.69·13-s + 1.36·19-s + 2/7·21-s + 25-s − 27-s + 1.48·31-s − 0.702·37-s − 1.69·39-s − 0.511·43-s − 0.918·49-s − 1.36·57-s − 1.21·61-s − 2/7·63-s + 1.82·67-s − 0.630·73-s − 75-s + 1.79·79-s + 81-s − 0.483·91-s − 1.48·93-s + 2/97·97-s − 1.88·103-s + 1.96·109-s + 0.702·111-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.159146285\)
\(L(\frac12)\) \(\approx\) \(1.159146285\)
\(L(2)\) 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 \( 1 + p T \)
good5 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 2 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - 22 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 26 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 46 T + p^{2} T^{2} \)
37 \( 1 + 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 74 T + p^{2} T^{2} \)
67 \( 1 - 122 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 + 46 T + p^{2} T^{2} \)
79 \( 1 - 142 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 2 T + p^{2} T^{2} \)
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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

−12.17598390036559572507728942182, −11.32781820773087690171169195152, −10.51807724963173039077624255474, −9.488884202479221612841582535349, −8.234364812269231408707988982858, −6.88141750819301618324524941498, −6.03494321582422021727699368452, −4.89789857048229982088640185305, −3.45490833041911840998630185916, −1.11266236078654724201175028387, 1.11266236078654724201175028387, 3.45490833041911840998630185916, 4.89789857048229982088640185305, 6.03494321582422021727699368452, 6.88141750819301618324524941498, 8.234364812269231408707988982858, 9.488884202479221612841582535349, 10.51807724963173039077624255474, 11.32781820773087690171169195152, 12.17598390036559572507728942182

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