Properties

Label 2-192-24.11-c1-0-2
Degree $2$
Conductor $192$
Sign $0.707 - 0.707i$
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  + 1.73i·3-s + 3.46·5-s − 2i·7-s − 2.99·9-s + 3.46i·11-s + 5.99i·15-s + 3.46·21-s + 6.99·25-s − 5.19i·27-s − 10.3·29-s − 10i·31-s − 5.99·33-s − 6.92i·35-s − 10.3·45-s + 3·49-s + ⋯
L(s)  = 1  + 0.999i·3-s + 1.54·5-s − 0.755i·7-s − 0.999·9-s + 1.04i·11-s + 1.54i·15-s + 0.755·21-s + 1.39·25-s − 0.999i·27-s − 1.92·29-s − 1.79i·31-s − 1.04·33-s − 1.17i·35-s − 1.54·45-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.707 - 0.707i$
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: \((2,\ 192,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29581 + 0.536744i\)
\(L(\frac12)\) \(\approx\) \(1.29581 + 0.536744i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
good5 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−12.90649354745569504346093811782, −11.42080890996111399603065516516, −10.40680032789476037594704772750, −9.761821418677728907157702231770, −9.135013810216059280878062108629, −7.53245001390505133599562759371, −6.16299848642427505324257100487, −5.17668557154051799532013434223, −3.96613534954855246776242677496, −2.18965243836546082585507017635, 1.68969402465132796719893245960, 2.93265545998427546509763475109, 5.50558532915946811707201965240, 5.94156164681195485811015626370, 7.10870659391611060195736760098, 8.574031530437562596343786013299, 9.191494158685320040968530216438, 10.50037454426579202494958929812, 11.55688822363957266631092405266, 12.61866298150081840950699032976

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