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Properties

Label 1-2183-2183.1014-r0-0-0
Degree $1$
Conductor $2183$
Sign $-0.997 - 0.0716i$
Analytic cond. $10.1378$
Root an. cond. $10.1378$
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  + (0.261 − 0.965i)2-s + (0.931 − 0.364i)3-s + (−0.863 − 0.505i)4-s + (0.777 − 0.628i)5-s + (−0.108 − 0.994i)6-s + (−0.392 + 0.919i)7-s + (−0.713 + 0.700i)8-s + (0.734 − 0.678i)9-s + (−0.403 − 0.915i)10-s + (−0.773 − 0.633i)11-s + (−0.987 − 0.155i)12-s + (−0.678 + 0.734i)13-s + (0.785 + 0.619i)14-s + (0.494 − 0.869i)15-s + (0.489 + 0.871i)16-s + (0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (0.261 − 0.965i)2-s + (0.931 − 0.364i)3-s + (−0.863 − 0.505i)4-s + (0.777 − 0.628i)5-s + (−0.108 − 0.994i)6-s + (−0.392 + 0.919i)7-s + (−0.713 + 0.700i)8-s + (0.734 − 0.678i)9-s + (−0.403 − 0.915i)10-s + (−0.773 − 0.633i)11-s + (−0.987 − 0.155i)12-s + (−0.678 + 0.734i)13-s + (0.785 + 0.619i)14-s + (0.494 − 0.869i)15-s + (0.489 + 0.871i)16-s + (0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.997 - 0.0716i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ -0.997 - 0.0716i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07877719131 - 2.195264453i\)
\(L(\frac12)\) \(\approx\) \(0.07877719131 - 2.195264453i\)
\(L(1)\) \(\approx\) \(1.019525250 - 1.102017791i\)
\(L(1)\) \(\approx\) \(1.019525250 - 1.102017791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.261 - 0.965i)T \)
3 \( 1 + (0.931 - 0.364i)T \)
5 \( 1 + (0.777 - 0.628i)T \)
7 \( 1 + (-0.392 + 0.919i)T \)
11 \( 1 + (-0.773 - 0.633i)T \)
13 \( 1 + (-0.678 + 0.734i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.993 + 0.114i)T \)
23 \( 1 + (0.738 - 0.674i)T \)
29 \( 1 + (-0.576 - 0.817i)T \)
31 \( 1 + (-0.687 + 0.725i)T \)
41 \( 1 + (0.0421 - 0.999i)T \)
43 \( 1 + (0.986 - 0.161i)T \)
47 \( 1 + (-0.874 - 0.484i)T \)
53 \( 1 + (0.897 - 0.441i)T \)
61 \( 1 + (0.820 - 0.571i)T \)
67 \( 1 + (-0.580 + 0.813i)T \)
71 \( 1 + (-0.324 + 0.945i)T \)
73 \( 1 + (-0.370 - 0.928i)T \)
79 \( 1 + (0.0240 - 0.999i)T \)
83 \( 1 + (-0.717 - 0.696i)T \)
89 \( 1 + (-0.0841 + 0.996i)T \)
97 \( 1 + (0.143 + 0.989i)T \)
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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

−20.1229296162690590648081474407, −19.418999462854880864119221664311, −18.47518367235667645317504830941, −17.86585126148083013457925041364, −17.12605493520750584632894562607, −16.45223898711806455546788997493, −15.56677533979448990464613568071, −14.92360375521943723412788406348, −14.50174859050932874008787936200, −13.68116658994796226107520303006, −13.03166690462854447409754041961, −12.74574575797973809866639053728, −11.02329321280504288393958420168, −10.16653510728004630476273906772, −9.75474012152065452164295401407, −9.07828091034329111344763913131, −7.85325546724056170527931020044, −7.46882284758242677541476560778, −6.88311407284615621597140769373, −5.71740284850401339471975734564, −5.08771196006330780784231306626, −4.12431007818349166449783686105, −3.266629529886180798211104964110, −2.72078210038993595776922534592, −1.3999783275099058477392969978, 0.56900538351794635163687051054, 1.68031906139084129835093380525, 2.49682035138730280595334284681, 2.85333189643247328679454273395, 3.90940328829077205293972477376, 5.077974735281679859490790316565, 5.46135036567953309628627137497, 6.54799865757039024888388783901, 7.62630889409086802345482458674, 8.71015242622435814631521957824, 9.080756322024434523477963175520, 9.6645181484076204543035997757, 10.37910545859509599268747547019, 11.64774872096298197977099976275, 12.15976586805002376458599139063, 13.00257847807028384008654307531, 13.35249785969666917182891239210, 14.18285330809100399390879171057, 14.650700276653042052206037337149, 15.7552257434993058704979782822, 16.40158280915902212744048878751, 17.60719653235440396293834105789, 18.29329543633697248821810231162, 18.832830032041787170749062028709, 19.34295168984379398297359342786

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