Properties

Degree 1
Conductor 43
Sign $0.988 + 0.150i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.222 − 0.974i)2-s + (−0.222 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + 6-s + 7-s + (0.623 + 0.781i)8-s + (−0.900 − 0.433i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)11-s + (−0.222 − 0.974i)12-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + ⋯
L(s,χ)  = 1  + (−0.222 − 0.974i)2-s + (−0.222 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + 6-s + 7-s + (0.623 + 0.781i)8-s + (−0.900 − 0.433i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)11-s + (−0.222 − 0.974i)12-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + (0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.988 + 0.150i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.988 + 0.150i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.988 + 0.150i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (4, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (0:\ ),\ 0.988 + 0.150i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7232886331 + 0.05489515438i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7232886331 + 0.05489515438i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8697486243 + 0.01451851518i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8697486243 + 0.01451851518i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−34.52681076372950365641120281537, −33.72317736423590674779825098859, −32.456324415346718961244931724, −31.22057818196709643320415781902, −30.011803656466189962684922295596, −28.4144662416231330042870927305, −27.78129828299117836123761746398, −25.83316248364788615455032205532, −25.122375659572952339113439935728, −23.91648576788700715642148710526, −23.403491925231846269308072264611, −21.53329384707526730636498123991, −19.96057946149252435222704496514, −18.19866781187993237899113722181, −17.71665110501897959666170054382, −16.53284192686535604496653600682, −14.866723499011531904814992076641, −13.52087216546020192570786518241, −12.614965936907460272953626819523, −10.51409414528628052995852420256, −8.56841861556557867292912747453, −7.761834711205413473288454115171, −6.011537228900623811306071466690, −4.99516283539441252084080144992, −1.52564474771815504295621599711, 2.38507933787459982222110127632, 4.076319329876355818151339979591, 5.62660769345534926518797703120, 8.21524445337110199068514123101, 9.71808967938563990302533229493, 10.76214497436558391025227115127, 11.59782537623479389776502755362, 13.68269849833816554990076551931, 14.70406725147150666989358653899, 16.568266654828461192097558308429, 17.87353866314983121474599543807, 18.797525306939581843562876208074, 20.83323168174463786388685670334, 21.14165910372931241628077443956, 22.31606439381315296100378605914, 23.519399061703099222047351806165, 25.80435298302628901877766666784, 26.65572090357067156367433349560, 27.6685910811618072602446731921, 28.75824987838509955332622455846, 29.85513258795787039211623042672, 31.06014200136010820953009783041, 32.15344797986469677278879498181, 33.7239064368591671332575654428, 34.28734914172099387468694770073

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