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Properties

Label 1-2183-2183.1025-r0-0-0
Degree $1$
Conductor $2183$
Sign $0.774 - 0.633i$
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  + (0.197 − 0.980i)2-s + (0.874 + 0.484i)3-s + (−0.922 − 0.386i)4-s + (−0.619 + 0.785i)5-s + (0.647 − 0.762i)6-s + (−0.0901 − 0.995i)7-s + (−0.561 + 0.827i)8-s + (0.530 + 0.847i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.619 − 0.785i)12-s + (0.530 − 0.847i)13-s + (−0.994 − 0.108i)14-s + (−0.922 + 0.386i)15-s + (0.700 + 0.713i)16-s + (−0.0901 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.197 − 0.980i)2-s + (0.874 + 0.484i)3-s + (−0.922 − 0.386i)4-s + (−0.619 + 0.785i)5-s + (0.647 − 0.762i)6-s + (−0.0901 − 0.995i)7-s + (−0.561 + 0.827i)8-s + (0.530 + 0.847i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.619 − 0.785i)12-s + (0.530 − 0.847i)13-s + (−0.994 − 0.108i)14-s + (−0.922 + 0.386i)15-s + (0.700 + 0.713i)16-s + (−0.0901 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.774 - 0.633i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ 0.774 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.856542623 - 0.6625477372i\)
\(L(\frac12)\) \(\approx\) \(1.856542623 - 0.6625477372i\)
\(L(1)\) \(\approx\) \(1.259747936 - 0.3687544642i\)
\(L(1)\) \(\approx\) \(1.259747936 - 0.3687544642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.197 - 0.980i)T \)
3 \( 1 + (0.874 + 0.484i)T \)
5 \( 1 + (-0.619 + 0.785i)T \)
7 \( 1 + (-0.0901 - 0.995i)T \)
11 \( 1 + (0.267 + 0.963i)T \)
13 \( 1 + (0.530 - 0.847i)T \)
17 \( 1 + (-0.0901 + 0.995i)T \)
19 \( 1 + (-0.302 - 0.953i)T \)
23 \( 1 + (-0.161 - 0.986i)T \)
29 \( 1 + (-0.947 - 0.319i)T \)
31 \( 1 + (0.976 - 0.214i)T \)
41 \( 1 + (0.935 - 0.353i)T \)
43 \( 1 + (0.267 - 0.963i)T \)
47 \( 1 + (-0.370 + 0.928i)T \)
53 \( 1 + (0.336 + 0.941i)T \)
61 \( 1 + (0.750 + 0.661i)T \)
67 \( 1 + (0.997 + 0.0721i)T \)
71 \( 1 + (0.989 - 0.143i)T \)
73 \( 1 + (-0.994 - 0.108i)T \)
79 \( 1 + (-0.0180 + 0.999i)T \)
83 \( 1 + (0.837 + 0.546i)T \)
89 \( 1 + (0.750 - 0.661i)T \)
97 \( 1 + (-0.994 + 0.108i)T \)
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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

−19.533396420014524921470233298845, −19.02714394431723248567917493111, −18.50236742949427885206601912499, −17.73398902402867102518397899983, −16.63142378979949996834306372949, −16.08665073567159681402017395546, −15.61027109586322836916416437907, −14.72192574462952023691876053168, −14.13812002636384325164376573023, −13.35680964208125518824085082609, −12.81980301379408519687813953445, −11.91551078611807000189767128160, −11.48095373324141253300591367931, −9.604529858168887140313239262156, −9.22591432889779513322437508152, −8.458514781349875820938825172517, −8.11748374520681436060650611171, −7.21080900933328311478812118776, −6.333051362960480964438221185267, −5.65053284047246044712963082197, −4.69938269940008815230424304719, −3.737264644480149250325631609543, −3.24845192464785600347262144238, −1.91178736092602380442978790204, −0.80936430163143164429453875251, 0.80272328399720560517721726266, 2.08049941871404173955868282793, 2.7480955915448531376151120719, 3.69095850238914321949771255731, 4.10392128498979592750311782679, 4.738257176677482379921553757250, 6.11773163549314822347804134945, 7.14582574187957639761451006325, 7.91876565356672674755928213935, 8.6091711094991879353088936766, 9.572227013341545686089481090853, 10.34670653248551965118055359993, 10.66240346721089977764746055550, 11.367368510467379621525320800303, 12.56357116676929436731332995473, 13.05389481039031735263703165002, 13.90081714903430427295633609594, 14.521712117684999145501333098233, 15.17937928340421245962626474874, 15.673940466214645847824818438422, 16.97295341064195285985211891495, 17.66164756164331496100365150132, 18.498462799035792836712315426811, 19.397128800029052892377342605498, 19.61787280695590709421889630286

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