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Properties

Label 1-2183-2183.26-r0-0-0
Degree $1$
Conductor $2183$
Sign $-0.417 + 0.908i$
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  + (0.700 + 0.713i)2-s + (−0.436 + 0.899i)3-s + (−0.0180 + 0.999i)4-s + (−0.891 + 0.452i)5-s + (−0.947 + 0.319i)6-s + (0.935 − 0.353i)7-s + (−0.725 + 0.687i)8-s + (−0.619 − 0.785i)9-s + (−0.947 − 0.319i)10-s + (0.468 − 0.883i)11-s + (−0.891 − 0.452i)12-s + (−0.619 + 0.785i)13-s + (0.907 + 0.419i)14-s + (−0.0180 − 0.999i)15-s + (−0.999 − 0.0361i)16-s + (0.935 + 0.353i)17-s + ⋯
L(s)  = 1  + (0.700 + 0.713i)2-s + (−0.436 + 0.899i)3-s + (−0.0180 + 0.999i)4-s + (−0.891 + 0.452i)5-s + (−0.947 + 0.319i)6-s + (0.935 − 0.353i)7-s + (−0.725 + 0.687i)8-s + (−0.619 − 0.785i)9-s + (−0.947 − 0.319i)10-s + (0.468 − 0.883i)11-s + (−0.891 − 0.452i)12-s + (−0.619 + 0.785i)13-s + (0.907 + 0.419i)14-s + (−0.0180 − 0.999i)15-s + (−0.999 − 0.0361i)16-s + (0.935 + 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.417 + 0.908i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ -0.417 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053803282 + 1.644649885i\)
\(L(\frac12)\) \(\approx\) \(1.053803282 + 1.644649885i\)
\(L(1)\) \(\approx\) \(0.9755787525 + 0.8577358515i\)
\(L(1)\) \(\approx\) \(0.9755787525 + 0.8577358515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.700 + 0.713i)T \)
3 \( 1 + (-0.436 + 0.899i)T \)
5 \( 1 + (-0.891 + 0.452i)T \)
7 \( 1 + (0.935 - 0.353i)T \)
11 \( 1 + (0.468 - 0.883i)T \)
13 \( 1 + (-0.619 + 0.785i)T \)
17 \( 1 + (0.935 + 0.353i)T \)
19 \( 1 + (0.336 - 0.941i)T \)
23 \( 1 + (0.796 - 0.605i)T \)
29 \( 1 + (0.267 + 0.963i)T \)
31 \( 1 + (0.647 - 0.762i)T \)
41 \( 1 + (0.126 - 0.992i)T \)
43 \( 1 + (0.468 + 0.883i)T \)
47 \( 1 + (0.0541 + 0.998i)T \)
53 \( 1 + (0.197 - 0.980i)T \)
61 \( 1 + (-0.968 + 0.250i)T \)
67 \( 1 + (0.958 + 0.284i)T \)
71 \( 1 + (0.837 - 0.546i)T \)
73 \( 1 + (0.907 + 0.419i)T \)
79 \( 1 + (0.997 + 0.0721i)T \)
83 \( 1 + (-0.674 + 0.738i)T \)
89 \( 1 + (-0.968 - 0.250i)T \)
97 \( 1 + (0.907 - 0.419i)T \)
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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

−19.674015418859990206009388643825, −18.858662410160015655498759564960, −18.33510151007236506524541070931, −17.44144118040410420409393796199, −16.83408533019273622746624297511, −15.64524867849613837056251120073, −15.03262278304263093048508570285, −14.35138765096909268566045914013, −13.60541620008961676080733231493, −12.588765865533203271452392497199, −12.1468974685461053291685593069, −11.85455064277478074742314226302, −11.045126850328080337859025418452, −10.18481780836478414596497702693, −9.24283792989530393048025372239, −8.1186121861518710608544751620, −7.62038720033511339995288334306, −6.77095545507407995737694456587, −5.58570111568077478729777100730, −5.14197776786960069382018765649, −4.42884179058993878982995626272, −3.35754627959622051223406062720, −2.42949544769045475606061263472, −1.455358469494517736392628025067, −0.85263087116811972505594278463, 0.80822600456949067922794287325, 2.661530728455782995593760585148, 3.43691288183896566060380616418, 4.19290521616540639807213018789, 4.74021606425464299317900262279, 5.47450311446809781635821578501, 6.50424234137380905077520215965, 7.107313693710290618783874501472, 8.02800399688078162341138391540, 8.676071518524237810653804369418, 9.54114551851897902283381867270, 10.876645989515476861897914351469, 11.170392363498766115184506735697, 11.88959339326509973448223575700, 12.5152881768420284841442693075, 13.90211381949876034248061387593, 14.36782996260346205778082293762, 14.88486301603902996174267703417, 15.58331821210247876710159327200, 16.37614684107272170874418342503, 16.84398055598135918339726718657, 17.4708166887520081678833281453, 18.38479434123789478603128014359, 19.331488796550406150947482723331, 20.15126502008660024376996773153

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