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Properties

Label 1-3192-3192.107-r1-0-0
Degree $1$
Conductor $3192$
Sign $-0.999 + 0.0345i$
Analytic cond. $343.028$
Root an. cond. $343.028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s − 53-s + (0.5 − 0.866i)55-s + (−0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + 5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s − 53-s + (0.5 − 0.866i)55-s + (−0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.999 + 0.0345i$
Analytic conductor: \(343.028\)
Root analytic conductor: \(343.028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3192,\ (1:\ ),\ -0.999 + 0.0345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001488999201 - 0.08626743916i\)
\(L(\frac12)\) \(\approx\) \(0.001488999201 - 0.08626743916i\)
\(L(1)\) \(\approx\) \(1.111449307 - 0.03837926176i\)
\(L(1)\) \(\approx\) \(1.111449307 - 0.03837926176i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good5 \( 1 + T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.912903094595700033483460552692, −18.46851534951913129835636114287, −17.66564196010624050040081736451, −16.98573318254274379771161920803, −16.61446970287790132528266941178, −15.6136586430007842035100521516, −14.78670516100580085362312215233, −14.19716346388659833234206401309, −13.665050032341821221807730422573, −12.845422609888427767484139549316, −11.98305357239849814615834193877, −11.63067485824306800936392381030, −10.31212386869620532580998810952, −9.98279643751614707139719793405, −9.26401415949878836277253238009, −8.61647308873185051404056135079, −7.504377383422828690846741632543, −6.835868500866671918267452749750, −6.23429361496228485561661377163, −5.27415233699397925551543250049, −4.649013029673930031924921410516, −3.80942055857375721734872360156, −2.59639515391148600133247427759, −2.06940598377376053801610401636, −1.20507905830766736088159106116, 0.0119698697455752431364885049, 1.26513801386806236818954247069, 1.768554398810805366172732523406, 3.06805911975214337102858747936, 3.40358031938606000833939324507, 4.71966633507666993611371947601, 5.43776263062300582655679097720, 6.07026513373216269300847267663, 6.715709796904427225597281613567, 7.76966708559593777585544279075, 8.4217724043358985444903071667, 9.26880282734356597588379921689, 9.878935179276343268054586810290, 10.62141020438708118233640134131, 11.19901963698611887106335737228, 12.316607774580128492494368084177, 12.75617167947252191255719213373, 13.58234337733918331257637034753, 14.2802623933751299578101164991, 14.7032405581855816137327088223, 15.6960921358600702545959734500, 16.469796447060488656052207604276, 17.04540652835683757613157771793, 17.80526839635733355144990409962, 18.15375888794991703076634208858

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