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Properties

Label 1-40-40.27-r0-0-0
Degree $1$
Conductor $40$
Sign $0.525 - 0.850i$
Analytic cond. $0.185759$
Root an. cond. $0.185759$
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  − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.185759\)
Root analytic conductor: \(0.185759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 40,\ (0:\ ),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7015832744 - 0.3911582944i\)
\(L(\frac12)\) \(\approx\) \(0.7015832744 - 0.3911582944i\)
\(L(1)\) \(\approx\) \(0.9119401687 - 0.3227714304i\)
\(L(1)\) \(\approx\) \(0.9119401687 - 0.3227714304i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

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

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