Properties

Label 1-6025-6025.413-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.722 - 0.691i$
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  + (0.544 + 0.838i)2-s + (0.777 − 0.629i)3-s + (−0.406 + 0.913i)4-s + (0.951 + 0.309i)6-s + (0.0261 + 0.999i)7-s + (−0.987 + 0.156i)8-s + (0.207 − 0.978i)9-s + (−0.824 − 0.566i)11-s + (0.258 + 0.965i)12-s + (−0.608 + 0.793i)13-s + (−0.824 + 0.566i)14-s + (−0.669 − 0.743i)16-s + (−0.760 + 0.649i)17-s + (0.933 − 0.358i)18-s + (0.0261 + 0.999i)19-s + ⋯
L(s)  = 1  + (0.544 + 0.838i)2-s + (0.777 − 0.629i)3-s + (−0.406 + 0.913i)4-s + (0.951 + 0.309i)6-s + (0.0261 + 0.999i)7-s + (−0.987 + 0.156i)8-s + (0.207 − 0.978i)9-s + (−0.824 − 0.566i)11-s + (0.258 + 0.965i)12-s + (−0.608 + 0.793i)13-s + (−0.824 + 0.566i)14-s + (−0.669 − 0.743i)16-s + (−0.760 + 0.649i)17-s + (0.933 − 0.358i)18-s + (0.0261 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109518175 - 0.4451677917i\)
\(L(\frac12)\) \(\approx\) \(1.109518175 - 0.4451677917i\)
\(L(1)\) \(\approx\) \(1.211255062 + 0.4311632564i\)
\(L(1)\) \(\approx\) \(1.211255062 + 0.4311632564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.544 + 0.838i)T \)
3 \( 1 + (0.777 - 0.629i)T \)
7 \( 1 + (0.0261 + 0.999i)T \)
11 \( 1 + (-0.824 - 0.566i)T \)
13 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (-0.760 + 0.649i)T \)
19 \( 1 + (0.0261 + 0.999i)T \)
23 \( 1 + (-0.233 - 0.972i)T \)
29 \( 1 + (-0.0523 + 0.998i)T \)
31 \( 1 + (-0.983 + 0.182i)T \)
37 \( 1 + (-0.0261 - 0.999i)T \)
41 \( 1 + (0.156 - 0.987i)T \)
43 \( 1 + (0.522 - 0.852i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (0.838 + 0.544i)T \)
59 \( 1 + (0.358 - 0.933i)T \)
61 \( 1 + (0.987 + 0.156i)T \)
67 \( 1 + (-0.544 - 0.838i)T \)
71 \( 1 + (-0.902 - 0.430i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.891 - 0.453i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.983 - 0.182i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−17.728545266276935252986408096375, −17.409213473871401620376766211379, −16.123872617280092752696658278329, −15.738200156934269699247239129266, −14.867678868059835112145804454174, −14.65621668445443370121262676988, −13.56318272193641831406105554351, −13.27672354210053940603081495611, −12.91075542681957198949920321487, −11.62895593061331376775583036530, −11.20178915435470341485507278188, −10.37381663743004379656099205639, −9.96059230338931551893331019221, −9.451746224922233242891476223622, −8.62687500556992870147964722342, −7.61556197403969450905743466771, −7.30506590892524446753096871206, −6.111964821396408445396079604435, −5.07547297619457401146331026391, −4.73164103993573467742896472606, −4.09443166660290837498929378852, −3.233782939941408324382770809660, −2.66480690158528004229074518493, −2.03939745375071354826400584227, −0.93984144291666598342149439570, 0.22348130720697462123060097354, 1.98233575810929781639831592551, 2.25583695347067063108680741405, 3.23780205522662212615653041399, 3.85546266424879644159015527443, 4.73278394955721402655121574204, 5.66332817461856871843791621346, 6.0317825093640650604319400463, 7.00544323462067103178462106278, 7.377152072856983326279081258248, 8.42581256766977543792778092274, 8.61261102129839090890617915994, 9.2156641508681222476662880059, 10.21934942555958586507890371136, 11.22831369094677090106796616594, 12.143069665415397241129127856685, 12.54519133098451336284062222359, 13.01237784306710703563615649389, 13.883408945982186478047642174765, 14.42866467032641004200624172650, 14.83639911537830957772799117691, 15.58088974864585938977408052048, 16.17784182701952177999828242038, 16.79634412229926068359680048910, 17.78352366840816234319311883847

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