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Properties

Label 1-43-43.41-r0-0-0
Degree $1$
Conductor $43$
Sign $0.785 - 0.618i$
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  + (0.623 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.785 - 0.618i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.785 - 0.618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080635913 - 0.3744212498i\)
\(L(\frac12)\) \(\approx\) \(1.080635913 - 0.3744212498i\)
\(L(1)\) \(\approx\) \(1.280882245 - 0.3436156609i\)
\(L(1)\) \(\approx\) \(1.280882245 - 0.3436156609i\)

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

−34.60783807993177042560117407090, −33.880040398853353564003601451147, −32.08829293407243624507372622198, −31.268548317995195168055875793853, −30.55880191381545369465640411481, −29.40016699976611126156239646135, −27.013831336271257381439597688188, −26.56942077453515726732048176123, −24.87767370504302811768120270409, −24.16236030438404606959737287728, −23.26973022608385730326248853262, −21.75965632891089159243766203612, −20.36501062618073871003270363532, −18.89306178301697212477294842940, −17.7809925473614684013623308487, −16.147591536531915809954363030035, −14.68914244848011822483290067656, −14.12986194182960007577513569085, −12.50991487466739020520927493603, −11.390928925825440596093034207744, −8.55938661068695429493767944327, −7.75497368268656056861401105254, −6.51347887663179263960441541998, −4.47641698915349259248497737323, −2.814364179001190979126476820031, 2.365548712664398481107437790367, 4.2272689498967395211299481211, 4.9309528474944841174778983311, 7.805569217865140738718807527999, 9.30553523234523564868766983318, 10.73865616624257370005959960931, 11.90956197609164568892122006304, 13.38993892792875414931871289158, 14.984601636608638010789420349224, 15.40453175976359157796210193568, 17.544272853317359765156080664979, 19.489732749269316862182510590685, 20.14573203176943567720645269980, 21.16032929911745847063033984388, 22.31705977267366379299651090311, 23.67261782132115401209791067711, 24.75791497227662504206511816293, 26.66666902483758736458065820805, 27.667213443066765276909747218437, 28.36653620880181195352597685128, 30.31217828659335215919364542781, 31.09311484796894637367637779067, 31.90473075068187642842021560498, 33.01789584370209512149087078498, 34.14103995492599513120274270577

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