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Properties

Label 1-6025-6025.1039-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.990 + 0.135i$
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  + (0.743 − 0.669i)2-s + (−0.866 + 0.5i)3-s + (0.104 − 0.994i)4-s + (−0.309 + 0.951i)6-s + (−0.838 + 0.544i)7-s + (−0.587 − 0.809i)8-s + (0.5 − 0.866i)9-s + (−0.998 − 0.0523i)11-s + (0.406 + 0.913i)12-s + (−0.777 + 0.629i)13-s + (−0.258 + 0.965i)14-s + (−0.978 − 0.207i)16-s + (−0.156 − 0.987i)17-s + (−0.207 − 0.978i)18-s + (−0.777 − 0.629i)19-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.866 + 0.5i)3-s + (0.104 − 0.994i)4-s + (−0.309 + 0.951i)6-s + (−0.838 + 0.544i)7-s + (−0.587 − 0.809i)8-s + (0.5 − 0.866i)9-s + (−0.998 − 0.0523i)11-s + (0.406 + 0.913i)12-s + (−0.777 + 0.629i)13-s + (−0.258 + 0.965i)14-s + (−0.978 − 0.207i)16-s + (−0.156 − 0.987i)17-s + (−0.207 − 0.978i)18-s + (−0.777 − 0.629i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.990 + 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4869478872 + 0.03306234889i\)
\(L(\frac12)\) \(\approx\) \(0.4869478872 + 0.03306234889i\)
\(L(1)\) \(\approx\) \(0.7171582158 - 0.2441019164i\)
\(L(1)\) \(\approx\) \(0.7171582158 - 0.2441019164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.838 + 0.544i)T \)
11 \( 1 + (-0.998 - 0.0523i)T \)
13 \( 1 + (-0.777 + 0.629i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.777 - 0.629i)T \)
23 \( 1 + (0.891 + 0.453i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (-0.933 - 0.358i)T \)
37 \( 1 + (0.998 + 0.0523i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.891 + 0.453i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (-0.743 + 0.669i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (-0.629 + 0.777i)T \)
73 \( 1 + (-0.156 + 0.987i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.0523 - 0.998i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−17.3877659176258570426664749842, −16.93129204422736564698260143242, −16.49022620680066211282880081796, −15.83241559170208570696263154183, −14.99094855775228340739057195391, −14.621548589675916712757791819502, −13.4048980058809209203293445386, −13.014149465636071924997888654727, −12.76190155081811572192994045880, −12.07387256174269052077657292751, −11.09111161701853931076645329605, −10.564268798446661832546956702567, −9.93711037509372727199100598116, −8.81771504398049484715915272035, −7.88868217824154661417560009333, −7.522884293212012457758400800693, −6.79099360250696439339265294043, −6.15172430191970531952581395983, −5.613469295094176891364024570004, −4.87066898337955879993403217836, −4.226485739536817293412273640745, −3.3091950755785600756170924224, −2.56967205981154021455469208306, −1.63673226495854039477841827426, −0.19957907493113293958730586925, 0.4672414520696746722586411228, 1.82551563229262417611434921660, 2.59219654136056376942468957890, 3.253941192371441372022911172097, 4.08284056934411616831589203389, 4.89590900758138222980691362754, 5.31477459272581417259150162242, 5.931710567322956198017168713622, 6.89083095953697820781525882179, 7.153096208291175511828111463340, 8.73028432706609113623032066807, 9.44559230792125563026775234230, 9.81844784019809893918922149448, 10.55122091243707743927492165447, 11.42088336806405832392201642623, 11.54331434013467743220802893065, 12.5438124451260350429150894785, 12.99588700844625743050131529657, 13.45294668730912716271094155986, 14.62301327929137605302339449492, 15.13664319025813106774769738579, 15.625444594322006634403197280786, 16.37720351549242321582254830392, 16.83718711277802841468678916022, 17.877877505182378276807950086159

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