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Properties

Label 1-2123-2123.1011-r1-0-0
Degree $1$
Conductor $2123$
Sign $0.999 + 0.0155i$
Analytic cond. $228.148$
Root an. cond. $228.148$
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  + (0.258 + 0.965i)2-s − i·3-s + (−0.866 + 0.5i)4-s + (0.991 + 0.130i)5-s + (0.965 − 0.258i)6-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s − 9-s + (0.130 + 0.991i)10-s + (0.5 + 0.866i)12-s + (−0.923 + 0.382i)13-s + (0.707 − 0.707i)14-s + (0.130 − 0.991i)15-s + (0.5 − 0.866i)16-s + (0.608 + 0.793i)17-s + (−0.258 − 0.965i)18-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s − i·3-s + (−0.866 + 0.5i)4-s + (0.991 + 0.130i)5-s + (0.965 − 0.258i)6-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s − 9-s + (0.130 + 0.991i)10-s + (0.5 + 0.866i)12-s + (−0.923 + 0.382i)13-s + (0.707 − 0.707i)14-s + (0.130 − 0.991i)15-s + (0.5 − 0.866i)16-s + (0.608 + 0.793i)17-s + (−0.258 − 0.965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2123\)    =    \(11 \cdot 193\)
Sign: $0.999 + 0.0155i$
Analytic conductor: \(228.148\)
Root analytic conductor: \(228.148\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2123} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2123,\ (1:\ ),\ 0.999 + 0.0155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.745740924 + 0.01356026307i\)
\(L(\frac12)\) \(\approx\) \(1.745740924 + 0.01356026307i\)
\(L(1)\) \(\approx\) \(1.045016702 + 0.1610916005i\)
\(L(1)\) \(\approx\) \(1.045016702 + 0.1610916005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
193 \( 1 \)
good2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.991 - 0.130i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (-0.608 - 0.793i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (-0.793 + 0.608i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (0.965 + 0.258i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
97 \( 1 + (-0.258 + 0.965i)T \)
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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

−19.74550153584984041529169479148, −19.13493616372540668147438386263, −18.139273856232081883446377532090, −17.65555124308785941663989890482, −16.742923223994224991663186464212, −16.04336407436514686297288647889, −15.04304621378444647221743912583, −14.45232970274433624254562274916, −13.919299065880971441175730623172, −12.80134068151380399697778836367, −12.369406033395153588025165279679, −11.56675063361945524904889959089, −10.523220889788154240121667454967, −10.10112597891457843217386099930, −9.44374615230026391763748788594, −8.93523226285676867632889728, −8.058497597860519364735413778612, −6.33720436440931175167809431251, −5.8305663903489076916971952316, −4.95537663331791962599664975298, −4.51421651009850977248738808042, −3.19589806894515778532837874946, −2.703601184568759534410483917, −1.98119561727082910151096571457, −0.54891174649790597196024649811, 0.44909118093446262378504341761, 1.561466000032744685736433803753, 2.561136777276444853245445478882, 3.54853539546225215810305276790, 4.552446439934210072413615400380, 5.54754883019274007267025858404, 6.25393365571956937939028329822, 6.72755532480806761072059975316, 7.496032724326182400811994493448, 8.16361455260515718301529660190, 9.136608362301011666679590238363, 9.86484373398033045366411487166, 10.62326400927985771145587968742, 11.93508592784303814731391497681, 12.619571285348666074252468426694, 13.31220538775675523716378586394, 13.70925765302690815121683115252, 14.53112644271459633996698310095, 14.93920362496590122702348843604, 16.35705688782666960942575055321, 16.80944921778611804360216896821, 17.499092584962177133486170543039, 17.80246742596709349969906611534, 18.95372692003689577278214340385, 19.28796823404876308025519455445

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