Properties

Degree 1
Conductor $ 2^{3} \cdot 5 $
Sign $0.525 + 0.850i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·3-s + i·7-s − 9-s + 11-s i·13-s i·17-s − 19-s − 21-s i·23-s i·27-s + 29-s − 31-s + i·33-s + i·37-s + 39-s + ⋯
L(s,χ)  = 1  + i·3-s + i·7-s − 9-s + 11-s i·13-s i·17-s − 19-s − 21-s i·23-s i·27-s + 29-s − 31-s + i·33-s + i·37-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.525 + 0.850i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.525 + 0.850i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(40\)    =    \(2^{3} \cdot 5\)
\( \varepsilon \)  =  $0.525 + 0.850i$
motivic weight  =  \(0\)
character  :  $\chi_{40} (3, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 40,\ (0:\ ),\ 0.525 + 0.850i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7015832744 + 0.3911582944i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7015832744 + 0.3911582944i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9119401687 + 0.3227714304i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9119401687 + 0.3227714304i\)

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

−35.122125834453898743831884437606, −33.80457709813138321773176703112, −32.58753197454238590740533921319, −31.12003428298199346923354335974, −30.0971994351986827050315741741, −29.28666097719713521204783741303, −27.883428506007810131250784550945, −26.38337483426292802936698400847, −25.269445915875329627552442127233, −23.92639158745413475851254561378, −23.21696771785133477076154526384, −21.593589323327567001176671697400, −19.887703187581400930829557858825, −19.176543805830934711741344483751, −17.54685466546225082919988802852, −16.71672928645412687732821755996, −14.57334623724062779575792727986, −13.56783189492185872773033389516, −12.22307268743900450537621130756, −10.868897953192585114440795293532, −8.940371155949579377574244484917, −7.37110915929680674050723007875, −6.26373631835534623225079304783, −3.96812450725671685803413562738, −1.64202537801390229418067778151, 2.89056781780750596559976448631, 4.67014511488030982111174235052, 6.14211875190494735953280392413, 8.45883954006282991693725964976, 9.57509793875861435371820853105, 11.054318087978597375813628502568, 12.39862134062951689437667133845, 14.38530492672731610600971896733, 15.36983053429381221235339712706, 16.57602095753138741705364509985, 17.96880266645786107177546621635, 19.583334729135430289489453237354, 20.82583463722073568508810719242, 22.013127749235026651765169384278, 22.83112681279834309789155808254, 24.776076615038713622041184822249, 25.67646987378714662053549730758, 27.280467228065419895394290647645, 27.83902827305057233268188607501, 29.20694324423191248769592152510, 30.76022116148276051216075375023, 31.9983823454100829952983009412, 32.76546716492736566693430288692, 34.14985841778778744419684566672, 34.98937625757406440227690267113

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