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Properties

Label 1-43-43.4-r0-0-0
Degree $1$
Conductor $43$
Sign $0.988 + 0.150i$
Analytic cond. $0.199691$
Root an. cond. $0.199691$
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  + (−0.222 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7232886331 + 0.05489515438i\)
\(L(\frac12)\) \(\approx\) \(0.7232886331 + 0.05489515438i\)
\(L(1)\) \(\approx\) \(0.8697486243 + 0.01451851518i\)
\(L(1)\) \(\approx\) \(0.8697486243 + 0.01451851518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−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