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Properties

Label 1-43-43.7-r1-0-0
Degree $1$
Conductor $43$
Sign $0.0861 - 0.996i$
Analytic cond. $4.62099$
Root an. cond. $4.62099$
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  − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.0861 - 0.996i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ 0.0861 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9197477990 - 0.8436709768i\)
\(L(\frac12)\) \(\approx\) \(0.9197477990 - 0.8436709768i\)
\(L(1)\) \(\approx\) \(0.8679828422 - 0.4058429752i\)
\(L(1)\) \(\approx\) \(0.8679828422 - 0.4058429752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−34.45786116028905241653470247152, −33.449959835310220127057985134049, −32.85033601298427520494306920868, −30.82964101792448072500199269662, −29.89805525780254720101460713335, −28.55913474612112704110269309246, −27.104367014013485234884802833643, −26.62873154441361867637989874986, −25.61523355235269918386877170955, −24.3377335335006632025507789482, −22.34312742129952092821767183731, −21.19768794114394319455197829469, −20.06701402255949239176438154445, −18.98047379402767906970900106782, −17.41119871258341444433738695709, −16.56181692370333606205782380282, −14.88582026183279639560637185018, −14.11651098976765992036488623780, −11.45698066352487407606972334062, −10.36911215794911280288166030177, −9.43800991223500845865315463337, −7.89943696573456148333166205595, −6.40702451959329461376804207481, −3.9370644710261553809570052198, −2.06110075350992274517754634234, 1.09580141148963236096486757852, 2.52271332353344334609617653779, 5.646701597283202001520821953513, 7.274221837677492256275699867459, 8.686153496405728458400500866573, 9.38259720648212090240080904773, 11.60641861797986386818867806824, 12.6126904840729971507181535187, 14.32354414208736381328855795403, 15.80672083209480138750179692690, 17.523393775315170123987799323931, 18.01182952892485477824263365019, 19.62436074727165960834586876179, 20.30192963838427358199968174424, 21.75834566481149942836236027872, 24.06758745889883642503268159709, 24.903929149817166522275972064274, 25.396383436861469299782498533293, 27.11634759005845422161916774886, 28.21226763417084744022917991808, 29.28580260602621801917013327816, 30.27256313198806360753900635578, 31.65968148928727128084356556708, 32.99439858440759570823697586390, 34.56341959616186880179420607754

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