Properties

Degree 1
Conductor 41
Sign $0.574 + 0.818i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (0.987 − 0.156i)6-s + (0.987 + 0.156i)7-s + (0.951 + 0.309i)8-s + i·9-s + (0.309 − 0.951i)10-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (0.156 + 0.987i)13-s + (−0.707 + 0.707i)14-s + (0.891 + 0.453i)15-s + (−0.809 + 0.587i)16-s + (0.453 + 0.891i)17-s + ⋯
L(s,χ)  = 1  + (−0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (0.987 − 0.156i)6-s + (0.987 + 0.156i)7-s + (0.951 + 0.309i)8-s + i·9-s + (0.309 − 0.951i)10-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (0.156 + 0.987i)13-s + (−0.707 + 0.707i)14-s + (0.891 + 0.453i)15-s + (−0.809 + 0.587i)16-s + (0.453 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.574 + 0.818i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.574 + 0.818i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.574 + 0.818i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (7, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (1:\ ),\ 0.574 + 0.818i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7082713911 + 0.3683166009i$
$L(\frac12,\chi)$  $\approx$  $0.7082713911 + 0.3683166009i$
$L(\chi,1)$  $\approx$  0.6492604909 + 0.1800369332i
$L(1,\chi)$  $\approx$  0.6492604909 + 0.1800369332i

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

−34.76535377343529378952241197816, −33.45682925357443850465254772751, −31.95533511379612402769874340062, −30.754526133926650718258408030349, −29.63085233663326145351185511017, −28.19838093088555062079053186955, −27.38081043952121661755464426366, −26.99996332379074288021013548555, −25.028631974672196351898850596711, −23.24775234965385191131218578412, −22.34768802777894655110935377447, −20.79049263444763193363731582071, −20.21217871349226842810955475520, −18.48455509827159246395746985908, −17.29636406033783365878457087858, −16.27496062657057060028403973087, −14.7323408743968200108172552308, −12.4034345518730169982483876286, −11.550039829245397726771828365236, −10.4849783547694281201224730305, −8.95494860296566188233624964836, −7.52438496617739594503614280728, −4.897493110075047331051250324478, −3.667300398002298216605119710108, −0.86832481979073651834535098322, 1.24944840551750639854101573138, 4.66650029731957742649648002246, 6.35801007816226432091884680398, 7.50386202187077269888968386518, 8.6943548691650040441729698391, 10.9348007862144358158335927028, 11.77883221610698536984779633919, 13.875871255353860970111877664694, 15.120649433879229812338075321286, 16.5698961319630421794512766731, 17.56072259798921367583591095216, 18.81644767764245730463400872660, 19.557912616636736826118384977260, 21.89339339990186514464558949004, 23.42467402328143611901726052923, 23.9385358989430791203271818031, 25.05080534087095879970021588633, 26.686682733545521104653226283246, 27.65100564120765146077939788134, 28.536704046244493231207498508182, 30.18770429969652737409323377430, 31.18298014528940332432363740175, 32.86439388851435587772204729846, 34.15715418802983523310918073219, 34.72798635802533357639873339899

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