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Properties

Label 1-2760-2760.2069-r0-0-0
Degree $1$
Conductor $2760$
Sign $1$
Analytic cond. $12.8173$
Root an. cond. $12.8173$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 11-s + 13-s − 17-s + 19-s + 29-s + 31-s − 37-s − 41-s − 43-s + 47-s + 49-s + 53-s + 59-s + 61-s − 67-s − 71-s − 73-s − 77-s − 79-s + 83-s + 89-s + 91-s + 97-s + ⋯
L(s)  = 1  + 7-s − 11-s + 13-s − 17-s + 19-s + 29-s + 31-s − 37-s − 41-s − 43-s + 47-s + 49-s + 53-s + 59-s + 61-s − 67-s − 71-s − 73-s − 77-s − 79-s + 83-s + 89-s + 91-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(12.8173\)
Root analytic conductor: \(12.8173\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2760} (2069, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2760,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.918356083\)
\(L(\frac12)\) \(\approx\) \(1.918356083\)
\(L(1)\) \(\approx\) \(1.216215938\)
\(L(1)\) \(\approx\) \(1.216215938\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−19.13720751095704568049756763560, −18.38742081611949947857254240944, −17.87573936278649284195169395520, −17.36322333581773280694133999155, −16.275151796355183324445520727502, −15.6647569609617165532797392283, −15.182220846289730423163460448196, −14.15155241449214739228233352474, −13.58014405335834479727381517074, −13.05830958070246133317934856240, −11.77839186106918754193795365697, −11.60386331358205476952416234511, −10.4572958800616419352085152999, −10.24362511983552591598186690674, −8.79058593140010591377003555679, −8.5391594252935714342012839467, −7.653922751879929769292503138318, −6.90903220414087570399295578845, −5.96175885634184789408460850431, −5.139730859161383606872409535612, −4.576848330897002542200224418920, −3.569302886811440170456609247, −2.6508243756832792929557590691, −1.78377887483777020184163188857, −0.82964837339369335479170319719, 0.82964837339369335479170319719, 1.78377887483777020184163188857, 2.6508243756832792929557590691, 3.569302886811440170456609247, 4.576848330897002542200224418920, 5.139730859161383606872409535612, 5.96175885634184789408460850431, 6.90903220414087570399295578845, 7.653922751879929769292503138318, 8.5391594252935714342012839467, 8.79058593140010591377003555679, 10.24362511983552591598186690674, 10.4572958800616419352085152999, 11.60386331358205476952416234511, 11.77839186106918754193795365697, 13.05830958070246133317934856240, 13.58014405335834479727381517074, 14.15155241449214739228233352474, 15.182220846289730423163460448196, 15.6647569609617165532797392283, 16.275151796355183324445520727502, 17.36322333581773280694133999155, 17.87573936278649284195169395520, 18.38742081611949947857254240944, 19.13720751095704568049756763560

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