Properties

Label 1-41-41.6-r1-0-0
Degree $1$
Conductor $41$
Sign $0.574 - 0.818i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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  + (−0.587 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ 0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7082713911 - 0.3683166009i\)
\(L(\frac12)\) \(\approx\) \(0.7082713911 - 0.3683166009i\)
\(L(1)\) \(\approx\) \(0.6492604909 - 0.1800369332i\)
\(L(1)\) \(\approx\) \(0.6492604909 - 0.1800369332i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.987 - 0.156i)T \)
11 \( 1 + (0.891 + 0.453i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.453 - 0.891i)T \)
19 \( 1 + (0.156 + 0.987i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.987 + 0.156i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (-0.891 + 0.453i)T \)
71 \( 1 + (-0.891 - 0.453i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.987 - 0.156i)T \)
97 \( 1 + (0.891 - 0.453i)T \)
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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

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

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