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Properties

Label 1-43-43.22-r1-0-0
Degree $1$
Conductor $43$
Sign $-0.863 - 0.503i$
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  + (−0.623 − 0.781i)2-s + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + 6-s − 7-s + (0.900 − 0.433i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)11-s + (−0.623 − 0.781i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s + (−0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + 6-s − 7-s + (0.900 − 0.433i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)11-s + (−0.623 − 0.781i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s + (−0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.863 - 0.503i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ -0.863 - 0.503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1054053054 - 0.3900837046i\)
\(L(\frac12)\) \(\approx\) \(0.1054053054 - 0.3900837046i\)
\(L(1)\) \(\approx\) \(0.4958056742 - 0.1921787255i\)
\(L(1)\) \(\approx\) \(0.4958056742 - 0.1921787255i\)

Euler product

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

−34.92536792258148701314569426534, −33.72960928722102665062275818000, −33.05864393992721237743127711601, −31.46030069972594153070404267625, −29.64714306926130996129436246175, −29.01493173255697605601444410012, −27.902458316237283635544390450097, −26.23126371977121243914640707754, −25.37173837210533963856873936405, −24.42517123594303822440225814080, −22.9577980006859578515331499926, −22.25287880498495567654870976688, −19.88649301482045938640359632710, −18.716261231664026383874665821733, −17.6819924922444588473096699976, −16.907169104453699209045057222245, −15.3458268222217986882155934803, −13.8124595074614612598439226790, −12.61245735566498458451960456473, −10.54329990297302720110742519835, −9.5490711137965876681680040380, −7.51379086614318658440245752914, −6.50959148719987269558665768959, −5.39194248130293473863232787345, −1.98297127705129717614369871831, 0.31180566149356018981619117229, 2.81555328701758415957416244905, 4.69301881409986213753699386326, 6.48388230773465929620484304193, 8.96694976561699129794887034542, 9.74600042762669450244381679292, 10.93791455220880097483534998869, 12.3652769717193273741510771318, 13.644576978521201121266744092102, 15.98493974903939487859238296959, 16.84494649510486752742849610887, 17.89866973599801925560993613713, 19.421353368288826773282091725477, 20.7324502335511626413725294173, 21.80737970417227387660387986782, 22.44939269611324043311414348206, 24.49052413504911601527412476861, 26.15738433794192657087987819016, 26.76498646845027129459169482019, 28.378507997918933770391704876796, 28.91720261488249095978811622513, 29.74789925566173855084644265489, 31.748361158531520564457942223679, 32.55301559437363267207950878134, 34.02072831906720117119188726072

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