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Properties

Label 1-43-43.36-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: $\chi_{43} (36, \cdot )$
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

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

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