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Properties

Label 1-2280-2280.2123-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.817 + 0.576i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)17-s + (0.984 + 0.173i)23-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.984 − 0.173i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)17-s + (0.984 + 0.173i)23-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.984 − 0.173i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (2123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.817 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287607357 + 0.4082223310i\)
\(L(\frac12)\) \(\approx\) \(1.287607357 + 0.4082223310i\)
\(L(1)\) \(\approx\) \(0.9946191993 + 0.04665999620i\)
\(L(1)\) \(\approx\) \(0.9946191993 + 0.04665999620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.642 - 0.766i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.342 - 0.939i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.642 + 0.766i)T \)
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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

−19.41600367217425930599123246248, −18.75917227478057690078423710976, −18.57830029271716480809489044436, −17.14036089158292714858272527256, −16.78576996005018836922906744293, −16.047825367673858067773004584524, −15.34310257318362487658022764627, −14.48810743008343598214040909145, −13.78005581141547027562637969480, −13.14781047826964407555656668173, −12.160304347090375216142208780802, −11.71309057075883429486213591788, −10.90645240735889579546682644901, −9.80421276189715077543239056170, −9.37244844281436292035104892032, −8.63340907714234665599115738758, −7.72307163199712989845755015441, −6.68113076489185966695159687497, −6.304637435935731483138561483869, −5.30194456792803738835415876549, −4.48868141368320927962046943792, −3.37636294892170195340660510339, −2.88279863561903952242382703923, −1.760282879013553400070116997516, −0.57082022303058964991297505598, 0.87979022159054658001060159403, 1.86466915869663959688906061235, 3.106254520953282385215082667557, 3.56849092386675488504789656774, 4.643805432132122759992005758066, 5.41809623205406655013321658153, 6.38714655129473757215888363357, 7.08252526701736633886157299691, 7.70595048655121741945428440326, 8.77350054101314755999379453449, 9.45646086134688279980119161060, 10.4183882573024090530864577226, 10.55799000422004156660748153415, 11.96290235300922265259199100869, 12.4945475099095032065584922087, 13.08556255546930282126977188667, 13.92800523683984121118218401920, 14.83816483023136800427614959756, 15.26110883615226704066747988254, 16.257530621009942348165537362598, 16.861173799224589632671668500139, 17.52961956176342730982316665276, 18.20226227206201893049611518706, 19.247662931413145289589001791531, 19.70136303268207203812620260960

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