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Properties

Label 1-6025-6025.117-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.175 + 0.984i$
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  + (0.987 + 0.156i)2-s + (0.156 + 0.987i)3-s + (0.951 + 0.309i)4-s + i·6-s + (0.972 − 0.233i)7-s + (0.891 + 0.453i)8-s + (−0.951 + 0.309i)9-s + (0.233 + 0.972i)11-s + (−0.156 + 0.987i)12-s + (0.522 − 0.852i)13-s + (0.996 − 0.0784i)14-s + (0.809 + 0.587i)16-s + (−0.972 − 0.233i)17-s + (−0.987 + 0.156i)18-s + (−0.0784 + 0.996i)19-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.156 + 0.987i)3-s + (0.951 + 0.309i)4-s + i·6-s + (0.972 − 0.233i)7-s + (0.891 + 0.453i)8-s + (−0.951 + 0.309i)9-s + (0.233 + 0.972i)11-s + (−0.156 + 0.987i)12-s + (0.522 − 0.852i)13-s + (0.996 − 0.0784i)14-s + (0.809 + 0.587i)16-s + (−0.972 − 0.233i)17-s + (−0.987 + 0.156i)18-s + (−0.0784 + 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.749418873 + 3.139969235i\)
\(L(\frac12)\) \(\approx\) \(3.749418873 + 3.139969235i\)
\(L(1)\) \(\approx\) \(2.179504764 + 1.036881909i\)
\(L(1)\) \(\approx\) \(2.179504764 + 1.036881909i\)

Euler product

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

−17.64638827737392325254809904304, −16.93208348467578756532097354296, −16.111818011099426937163783582446, −15.42867767323100142485920755259, −14.727612203312408716233956038883, −14.08758502798211548418620747399, −13.58649785509985009398513765689, −13.21593392571082858680063429151, −12.30386341579108147124589866518, −11.53504536128609933831984852688, −11.350555391025636646027819049601, −10.77038865220800958385472459464, −9.474644572735838129260456434925, −8.63872230980536448537765559091, −8.18820469953852608938505918016, −7.30369053851781838960166402700, −6.52183451078552887543357615278, −6.24366112250651353271445476512, −5.27768921463033427274065502532, −4.66428498904396529743409256050, −3.813532949549317197978632327447, −2.95583544931998947390910855276, −2.3120167475607694151505118811, −1.52224272125895656914525221827, −0.94149511288825591623430682527, 1.04202976694411459006773844510, 2.27045075141467544879908065899, 2.57015436545915896082131281069, 3.83620301708285715908375537228, 4.16788616459479254528274573600, 4.7565287491193968267264578183, 5.48969013278525012953526970587, 6.11004588828893643512662924386, 7.02980265740509605184235433360, 7.848126090442684145151111429855, 8.34782857448267665812422886654, 9.17650807122743911289133085765, 10.20503226819914298634514560274, 10.671318756409514677359325800440, 11.16467531236161242736333377344, 12.022525976259363523191408486711, 12.53024573986452104745704937513, 13.49642359187113785875826405388, 14.02489939374816020108511636574, 14.775749525264363953165886597660, 15.020765953222266049345888848414, 15.7411544772746011762643777680, 16.3209646417619484935186821291, 17.0972858654799154704827654548, 17.60627245296725522141326453789

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