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Properties

Label 1-2183-2183.110-r0-0-0
Degree $1$
Conductor $2183$
Sign $0.0148 - 0.999i$
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  + (0.947 + 0.319i)2-s + (−0.856 − 0.515i)3-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (−0.647 − 0.762i)6-s + (0.907 − 0.419i)7-s + (0.561 + 0.827i)8-s + (0.468 + 0.883i)9-s + (0.647 − 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.370 − 0.928i)12-s + (−0.468 + 0.883i)13-s + (0.994 − 0.108i)14-s + (−0.796 + 0.605i)15-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + ⋯
L(s)  = 1  + (0.947 + 0.319i)2-s + (−0.856 − 0.515i)3-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (−0.647 − 0.762i)6-s + (0.907 − 0.419i)7-s + (0.561 + 0.827i)8-s + (0.468 + 0.883i)9-s + (0.647 − 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.370 − 0.928i)12-s + (−0.468 + 0.883i)13-s + (0.994 − 0.108i)14-s + (−0.796 + 0.605i)15-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.0148 - 0.999i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ 0.0148 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.664469024 - 1.639906883i\)
\(L(\frac12)\) \(\approx\) \(1.664469024 - 1.639906883i\)
\(L(1)\) \(\approx\) \(1.529958585 - 0.3917793443i\)
\(L(1)\) \(\approx\) \(1.529958585 - 0.3917793443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.947 + 0.319i)T \)
3 \( 1 + (-0.856 - 0.515i)T \)
5 \( 1 + (0.370 - 0.928i)T \)
7 \( 1 + (0.907 - 0.419i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (-0.468 + 0.883i)T \)
17 \( 1 + (-0.907 - 0.419i)T \)
19 \( 1 + (-0.976 + 0.214i)T \)
23 \( 1 + (0.161 - 0.986i)T \)
29 \( 1 + (0.947 - 0.319i)T \)
31 \( 1 + (-0.976 - 0.214i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (-0.267 - 0.963i)T \)
47 \( 1 + (-0.370 - 0.928i)T \)
53 \( 1 + (0.647 + 0.762i)T \)
61 \( 1 + (0.947 + 0.319i)T \)
67 \( 1 + (-0.561 - 0.827i)T \)
71 \( 1 + (-0.370 - 0.928i)T \)
73 \( 1 + (-0.994 + 0.108i)T \)
79 \( 1 + (0.856 - 0.515i)T \)
83 \( 1 + (0.0541 + 0.998i)T \)
89 \( 1 + (0.947 - 0.319i)T \)
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

−20.06470351936194248791677819238, −19.41098209820504000336659118432, −18.29793251704393768883594831890, −17.63259731948594331995954893831, −17.34286708891559118302235229906, −16.04577549959382496874222824898, −15.30016606449216329189979922301, −14.80565036965500808249333434909, −14.50173843796745960396288191678, −13.147495949273447877491995590688, −12.69715859798512853056383344001, −11.6685616098654314987859849776, −11.30538317583072586382473554691, −10.49470117246534906252234349910, −10.06965539113821166930405881818, −9.10660729169174725746127674751, −7.71852699576121098066556094892, −6.86481377525769957493507956897, −6.2733360663574847052678189788, −5.43388479313264201468205926575, −4.80500522923768730853175309500, −4.11807715663314790856539574580, −3.0926206212264050477199883671, −2.184575342176535696257048557975, −1.4084475926388172559715722032, 0.58351083655271099939817078267, 1.81117822360305279140490526130, 2.26862634537888716389287329456, 3.96144735178578393080038811361, 4.573324082418823758579270915083, 5.1188035676939790471600942372, 5.96110136952281270043221841329, 6.645682519131859505463307553752, 7.349255947723490868954157374144, 8.37388149781537075153158237580, 8.84790684665549676175760230418, 10.41098878696524391323264597888, 10.977789131948829006196409790127, 11.84036078405710672803672813453, 12.17363439935774010552748322945, 13.16090595063136136386025123371, 13.662758002862978061572193365167, 14.23656595099680107095918763417, 15.18601477486448595188895249786, 16.30276732571929242173491327959, 16.60136029794591116991688573505, 17.202880852450806981699210960006, 17.807984482956316509095693734020, 18.84270906897996454000124238133, 19.71651256162120985441065292697

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