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Properties

Label 1-6025-6025.5033-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: $\chi_{6025} (5033, \cdot )$
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

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

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