Properties

Label 1-43-43.6-r0-0-0
Degree $1$
Conductor $43$
Sign $0.736 - 0.675i$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.736 - 0.675i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.736 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062973404 - 0.4136849104i\)
\(L(\frac12)\) \(\approx\) \(1.062973404 - 0.4136849104i\)
\(L(1)\) \(\approx\) \(1.281795694 - 0.3423057538i\)
\(L(1)\) \(\approx\) \(1.281795694 - 0.3423057538i\)

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

−34.35123470531189341937925306703, −33.4637713257218731271959355403, −32.52541454916462303596608896802, −31.50609852754867947086485653936, −29.97627395655533468200699306087, −29.37771016219550144226377426941, −27.60630021380698140569717651773, −26.61154235954059879888363960619, −25.27212788247672027739344399431, −23.53722891701169444596997070874, −22.64000907782534067761048444312, −22.10511635612365401816307500844, −20.504326387867857747071701170656, −19.52794369926892615390476237257, −17.340269016533038665614393300016, −16.1119927419996294217674625963, −15.055023760673684597183648077760, −13.98033703190551588163770949528, −12.15769709370406409158855491779, −11.03135786332682597247967294348, −9.982617212940609794290545511925, −7.3036800779324419824399179562, −6.05798551481637015006045765322, −4.26946752696245929219109306146, −3.28094717712705603898194001140, 2.04207198603197446195261156079, 4.297617203734646786314182174756, 5.827306809281636210352851808801, 7.00734058174388255700433727443, 8.80248953995798203098224767936, 11.3101142053880267348294074073, 12.28507179147785800679626880372, 12.99190428180432611567132771767, 14.611452038236319763941221613116, 16.122120451661042992355717557479, 17.11687792680087581462770218955, 19.14803993896234736559989673004, 19.89264453192944600924673808152, 21.69959556284112712599834078935, 22.58471435844809456801338095961, 24.04514573055887535627535659919, 24.421364197778380215681194164244, 25.7092158583536843989574562930, 28.07162598900808192462247008189, 28.72033461081603112326729915245, 29.97767583802619608376648309752, 31.09649627625923147597892870936, 31.96130395439532110985251468270, 33.18101304050135073666329332679, 34.737198939494816839854463184393

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