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  :  $\chi_{43} (11, \cdot )$
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

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

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