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Properties

Label 1-6025-6025.4837-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.458 - 0.888i$
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  + (−0.453 + 0.891i)2-s + (−0.453 + 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (−0.233 − 0.972i)7-s + (0.987 − 0.156i)8-s + (−0.587 − 0.809i)9-s + (0.0784 − 0.996i)11-s + (0.987 − 0.156i)12-s + (−0.233 + 0.972i)13-s + (0.972 + 0.233i)14-s + (−0.309 + 0.951i)16-s + (0.382 + 0.923i)17-s + (0.987 − 0.156i)18-s + (−0.852 + 0.522i)19-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (−0.453 + 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (−0.233 − 0.972i)7-s + (0.987 − 0.156i)8-s + (−0.587 − 0.809i)9-s + (0.0784 − 0.996i)11-s + (0.987 − 0.156i)12-s + (−0.233 + 0.972i)13-s + (0.972 + 0.233i)14-s + (−0.309 + 0.951i)16-s + (0.382 + 0.923i)17-s + (0.987 − 0.156i)18-s + (−0.852 + 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.458 - 0.888i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1010562412 + 0.1659248018i\)
\(L(\frac12)\) \(\approx\) \(-0.1010562412 + 0.1659248018i\)
\(L(1)\) \(\approx\) \(0.4808842606 + 0.3170241833i\)
\(L(1)\) \(\approx\) \(0.4808842606 + 0.3170241833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 + (-0.453 + 0.891i)T \)
7 \( 1 + (-0.233 - 0.972i)T \)
11 \( 1 + (0.0784 - 0.996i)T \)
13 \( 1 + (-0.233 + 0.972i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.852 + 0.522i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.891 - 0.453i)T \)
43 \( 1 + (0.972 + 0.233i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.233 - 0.972i)T \)
79 \( 1 + (0.987 - 0.156i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.649 + 0.760i)T \)
97 \( 1 + T \)
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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

−17.6115028347114675313728468212, −16.82102003294797118552550058489, −16.24072292633041294421206205544, −15.238150741911777005718797985961, −14.6395785930560482827634775334, −13.64704176089424986368775868051, −12.977924642885562341006066030524, −12.60916350643454553566831758434, −11.88911017523362861999821112544, −11.633404049945905110109197576396, −10.62261589796520997777222744873, −10.0585493809593467902011967781, −9.373626427083287576969629066518, −8.47731407160406815920216907652, −8.08745412501700166144639331750, −7.20825118142232903926549614912, −6.63341144281364373178187482662, −5.64142650301019086005639573901, −5.00289213151995932082121740182, −4.27525483354937780748546239655, −3.03406521407263423113221235914, −2.466704041932796169666151277854, −2.02217585149876090087597257643, −0.94454230507570915393406166956, −0.08267735602488053111102019042, 0.974683779918044076578247883971, 1.867963956273266088869298161457, 3.500249933226978378036001554498, 3.81396088162635536321516471568, 4.61629887400667308895023090495, 5.332406830246960357975362604318, 6.24136832686059715433750418847, 6.40452768881380786836135838058, 7.45292835856281786234595202153, 8.09521248466829440868848630480, 9.00633993017320453109369220268, 9.321357019656853970045869782, 10.30175181082661996028719514437, 10.62029510026077145909907319602, 11.205280644524167861266127012274, 12.19289454277406913965684403652, 13.0340207868797355896508081298, 13.97494155798838699741271080292, 14.29742713788229675998139846589, 14.92785980755025844453641799901, 15.80194020853513042769226109298, 16.37559071936396205044753838392, 16.67465726132272712748029763962, 17.25244593772232428031748139497, 17.848802362043822285665198493955

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