Properties

Degree 1
Conductor 43
Sign $0.785 + 0.618i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.785 + 0.618i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (21, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 43,\ (0:\ ),\ 0.785 + 0.618i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.080635913 + 0.3744212498i$
$L(\frac12,\chi)$  $\approx$  $1.080635913 + 0.3744212498i$
$L(\chi,1)$  $\approx$  1.280882245 + 0.3436156609i
$L(1,\chi)$  $\approx$  1.280882245 + 0.3436156609i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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