Properties

Label 1-6025-6025.4622-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.898 - 0.439i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.156 + 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (0.522 + 0.852i)7-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.852 + 0.522i)11-s + (−0.987 + 0.156i)12-s + (0.972 + 0.233i)13-s + (−0.760 + 0.649i)14-s + (0.809 − 0.587i)16-s + (−0.522 + 0.852i)17-s + (−0.156 + 0.987i)18-s + (0.649 − 0.760i)19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (0.522 + 0.852i)7-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.852 + 0.522i)11-s + (−0.987 + 0.156i)12-s + (0.972 + 0.233i)13-s + (−0.760 + 0.649i)14-s + (0.809 − 0.587i)16-s + (−0.522 + 0.852i)17-s + (−0.156 + 0.987i)18-s + (0.649 − 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4903032010 + 2.119318779i\)
\(L(\frac12)\) \(\approx\) \(-0.4903032010 + 2.119318779i\)
\(L(1)\) \(\approx\) \(0.9425201391 + 1.071595699i\)
\(L(1)\) \(\approx\) \(0.9425201391 + 1.071595699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
7 \( 1 + (0.522 + 0.852i)T \)
11 \( 1 + (-0.852 + 0.522i)T \)
13 \( 1 + (0.972 + 0.233i)T \)
17 \( 1 + (-0.522 + 0.852i)T \)
19 \( 1 + (0.649 - 0.760i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (-0.852 - 0.522i)T \)
37 \( 1 + (-0.0784 + 0.996i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
47 \( 1 + (-0.987 + 0.156i)T \)
53 \( 1 + (0.156 + 0.987i)T \)
59 \( 1 + (-0.891 - 0.453i)T \)
61 \( 1 + (0.891 - 0.453i)T \)
67 \( 1 + (-0.987 - 0.156i)T \)
71 \( 1 + (-0.233 + 0.972i)T \)
73 \( 1 + (0.972 - 0.233i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.233 + 0.972i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−17.89801647568921232878886867255, −16.52960508538270199461803962433, −16.05488924956828672428992662669, −15.17185689465552801895713654048, −14.279347177954579072689148662445, −14.07257546086531753568692768630, −13.22989184897996922856171437081, −13.08817740092810791386865197354, −12.05665721732942460796207985383, −11.29646465043688720470542022837, −10.7120657516475072023994470536, −10.090273944871316711486706259590, −9.47785998125218696715916530289, −8.60107907711561580313459651873, −8.10713514332322507235726689034, −7.59568467431841339645346523587, −6.546834113296290743844357601245, −5.61031224431543692837776696666, −4.77744218924031907175524002518, −4.072907340135111315229135172442, −3.472145298182551820550895085500, −2.814955508767247707160229138681, −2.006755171479394615876451778618, −1.28793083811592076449861584656, −0.43320030849888551101325396234, 1.385663509808493102961200549750, 2.1097262311336519886282875971, 3.08193992047960397369481684815, 3.724835741548052289944576242413, 4.57774384224293736757967584235, 5.145540898302650235028102019043, 5.89070893565831024184075343353, 6.72363743470818566430445041440, 7.51407659629998788385879201895, 8.03286202156099590601453035746, 8.68839740091323190567761625951, 9.11229757034561254526909030502, 9.84355103573577512849575299815, 10.669827634877053543950105194729, 11.5394181447830605925157109094, 12.54134066267921584634071462631, 13.01692484829083613865795306762, 13.6796831971633513721102498013, 14.20649279635640314898896234831, 15.04378384640923860786085992238, 15.407847935589244198365261010953, 15.81310279321953723831895268254, 16.52316295487603542271698721020, 17.57145091982607039747861545505, 18.109880784904839555359959845522

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