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Properties

Label 2-192-24.5-c2-0-5
Degree $2$
Conductor $192$
Sign $0.258 - 0.965i$
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  + 3i·3-s + 13.8·7-s − 9·9-s + 13.8i·13-s + 26i·19-s + 41.5i·21-s − 25·25-s − 27i·27-s + 41.5·31-s − 69.2i·37-s − 41.5·39-s + 22i·43-s + 142.·49-s − 78·57-s − 96.9i·61-s + ⋯
L(s)  = 1  + i·3-s + 1.97·7-s − 9-s + 1.06i·13-s + 1.36i·19-s + 1.97i·21-s − 25-s − i·27-s + 1.34·31-s − 1.87i·37-s − 1.06·39-s + 0.511i·43-s + 2.91·49-s − 1.36·57-s − 1.59i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ 0.258 - 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.35544 + 1.04007i\)
\(L(\frac12)\) \(\approx\) \(1.35544 + 1.04007i\)
\(L(2)\) 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 \( 1 - 3iT \)
good5 \( 1 + 25T^{2} \)
7 \( 1 - 13.8T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 26iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 - 41.5T + 961T^{2} \)
37 \( 1 + 69.2iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 + 96.9iT - 3.72e3T^{2} \)
67 \( 1 + 122iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + 69.2T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 2T + 9.40e3T^{2} \)
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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

−12.10807436963661726464233800867, −11.42816634854329914954733567798, −10.66418932780235577695537300132, −9.568002675331722900875314647745, −8.487689019006686538968344785361, −7.72169551085210224211095598627, −5.92673665926523798684845191752, −4.81063456242074945471430763214, −3.97464103087090983432358712299, −1.93055013428607646341403966750, 1.14897359751057920284244212544, 2.59239956566227838793339870495, 4.68613849022063594021266748155, 5.70790755682818942747638374631, 7.14586740560967380163671535738, 8.047549360301403423062826550248, 8.638741939753299542273919825664, 10.35423500216452738715855348937, 11.46047560363475819958697374411, 11.86144979600781802884024637164

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