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Properties

Label 1-6025-6025.4284-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.451 + 0.892i$
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  + (−0.669 + 0.743i)2-s + (0.978 + 0.207i)3-s + (−0.104 − 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.913 + 0.406i)7-s + (0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.669 + 0.743i)11-s + (0.104 − 0.994i)12-s + (0.913 − 0.406i)13-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + 17-s + (−0.913 + 0.406i)18-s + (0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.978 + 0.207i)3-s + (−0.104 − 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.913 + 0.406i)7-s + (0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.669 + 0.743i)11-s + (0.104 − 0.994i)12-s + (0.913 − 0.406i)13-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + 17-s + (−0.913 + 0.406i)18-s + (0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.156913483 + 1.326642214i\)
\(L(\frac12)\) \(\approx\) \(2.156913483 + 1.326642214i\)
\(L(1)\) \(\approx\) \(1.232098311 + 0.5180889452i\)
\(L(1)\) \(\approx\) \(1.232098311 + 0.5180889452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−17.93140513543027911106937082003, −16.853370995017245346523607744630, −16.556730890745778573247827904806, −15.67757114125675889873763937132, −14.88292280923101735508863697965, −14.122386607318468620035560141491, −13.614995547493050374776908685181, −13.06803425897423125522026685380, −12.28028906283045252809594732162, −11.54988764651020933963049373014, −10.88666815517822774376022908417, −10.29556567501985363239048199227, −9.64083067640413176775467708165, −8.698626629963720502610425792978, −8.434132544910599679346223021119, −7.72022960647880826505435218167, −7.28811555637016777206176975109, −6.256147169688621319283663304166, −5.17694319074060831192903377422, −4.32506631817346747502139001832, −3.46233253440488637516690244372, −3.19708118541595668559308138916, −2.12741973979026907620045496971, −1.42897318046772220340675741813, −0.91407464647325577439319725089, 0.9127334816916904413640223383, 1.68872688404416575118980989776, 2.414912254936979814630843559951, 3.26492437134885273001324913504, 4.31406760644258871782212570354, 5.096867320209822215392593890133, 5.427649393535805670265892630227, 6.62308415689051237715583233011, 7.28924432668809526412479425666, 7.9265978368135986893152210883, 8.408466069979076080329201449393, 8.96198546559813183829479761887, 9.74119832834465867483019726072, 10.287712195444758033179486911762, 11.00613457390059617168832507, 11.69890737549793433226205918879, 12.96440548249899979963086606913, 13.26188364534255379811710357138, 14.3018229105422270552109329125, 14.58028989286762355910977830804, 15.40284030497335256825472742916, 15.61746133136436739201648031198, 16.354666844555493252395190054416, 17.345972013791922367844259855476, 17.79763374005711266069922882743

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