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Properties

Label 1-6025-6025.27-r1-0-0
Degree $1$
Conductor $6025$
Sign $-0.894 - 0.446i$
Analytic cond. $647.476$
Root an. cond. $647.476$
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  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.987 − 0.156i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.453 − 0.891i)11-s + 12-s + (0.707 − 0.707i)13-s + (0.453 − 0.891i)14-s + (0.309 + 0.951i)16-s + (0.453 + 0.891i)17-s + (0.809 + 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.987 − 0.156i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.453 − 0.891i)11-s + 12-s + (0.707 − 0.707i)13-s + (0.453 − 0.891i)14-s + (0.309 + 0.951i)16-s + (0.453 + 0.891i)17-s + (0.809 + 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(647.476\)
Root analytic conductor: \(647.476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (1:\ ),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1326816692 + 0.5625582025i\)
\(L(\frac12)\) \(\approx\) \(-0.1326816692 + 0.5625582025i\)
\(L(1)\) \(\approx\) \(0.4751074163 + 0.3207922937i\)
\(L(1)\) \(\approx\) \(0.4751074163 + 0.3207922937i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.987 - 0.156i)T \)
11 \( 1 + (-0.453 - 0.891i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.453 + 0.891i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (0.156 + 0.987i)T \)
29 \( 1 + (-0.951 + 0.309i)T \)
31 \( 1 + (0.891 - 0.453i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.987 - 0.156i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.891 + 0.453i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−17.304977530786215125643266892805, −16.66964178441297248302934367479, −16.09729834900586408629739646166, −15.425258447713040484619017978391, −14.21741733494189878632866831180, −13.570269746308875372784913608820, −12.98723860050597685771759309450, −12.52874620074110162973622649698, −11.87308733146592812218494166862, −11.31124616150118786869048334649, −10.59346726344960150618668558985, −9.98330777673001709882734278038, −9.33978349623726331753363891937, −8.644088644920660739678548000370, −7.74052165817131902901187355569, −7.00076005474030094238108428401, −6.525805984925569092884362716806, −5.50493656138763385280882463548, −4.80022334479196757158698138564, −4.16947493503572310681176145751, −3.12056022292367457226201867369, −2.43765724827336704302709975438, −1.76497114514595358569240695422, −0.74107405649225007992076514123, −0.18879465847778359460072585859, 0.71637549311970415817675801551, 1.37024376396513799424762035681, 3.0384478611690723570484750969, 3.76263109016874960421497098666, 4.21137701525953289398141873196, 5.49868374367208665943166055595, 5.86584827144182804245457841937, 6.05324434494238138538382617251, 7.18008419984402311548935590464, 7.731553222904637408352795119635, 8.70133898169665667505615947334, 9.1492911596593711342210770412, 10.13443660312547641122796554894, 10.393365644137059754518914975592, 11.00223468666135577736475885531, 12.05853474515026440554727621687, 12.83675566116392316553543627048, 13.30880633551788810516249544230, 14.10317826589709639699728980867, 14.935324442611958849186164961646, 15.607471642106722815607820560913, 15.985408469037160529646710975, 16.51494441582472132092446917601, 17.19927346986629582976995601184, 17.59599141408423228806077191805

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