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Properties

Label 1-6025-6025.1153-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0158 - 0.999i$
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  + (−0.777 + 0.629i)2-s + (−0.0523 + 0.998i)3-s + (0.207 − 0.978i)4-s + (−0.587 − 0.809i)6-s + (0.983 + 0.182i)7-s + (0.453 + 0.891i)8-s + (−0.994 − 0.104i)9-s + (−0.878 − 0.477i)11-s + (0.965 + 0.258i)12-s + (0.130 − 0.991i)13-s + (−0.878 + 0.477i)14-s + (−0.913 − 0.406i)16-s + (−0.972 − 0.233i)17-s + (0.838 − 0.544i)18-s + (0.983 + 0.182i)19-s + ⋯
L(s)  = 1  + (−0.777 + 0.629i)2-s + (−0.0523 + 0.998i)3-s + (0.207 − 0.978i)4-s + (−0.587 − 0.809i)6-s + (0.983 + 0.182i)7-s + (0.453 + 0.891i)8-s + (−0.994 − 0.104i)9-s + (−0.878 − 0.477i)11-s + (0.965 + 0.258i)12-s + (0.130 − 0.991i)13-s + (−0.878 + 0.477i)14-s + (−0.913 − 0.406i)16-s + (−0.972 − 0.233i)17-s + (0.838 − 0.544i)18-s + (0.983 + 0.182i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0158 - 0.999i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0158 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2516380719 - 0.2556671109i\)
\(L(\frac12)\) \(\approx\) \(0.2516380719 - 0.2556671109i\)
\(L(1)\) \(\approx\) \(0.6155171606 + 0.2295209506i\)
\(L(1)\) \(\approx\) \(0.6155171606 + 0.2295209506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.777 + 0.629i)T \)
3 \( 1 + (-0.0523 + 0.998i)T \)
7 \( 1 + (0.983 + 0.182i)T \)
11 \( 1 + (-0.878 - 0.477i)T \)
13 \( 1 + (0.130 - 0.991i)T \)
17 \( 1 + (-0.972 - 0.233i)T \)
19 \( 1 + (0.983 + 0.182i)T \)
23 \( 1 + (0.0784 - 0.996i)T \)
29 \( 1 + (-0.358 - 0.933i)T \)
31 \( 1 + (0.958 - 0.284i)T \)
37 \( 1 + (-0.983 - 0.182i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (-0.760 + 0.649i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (0.629 - 0.777i)T \)
59 \( 1 + (0.544 - 0.838i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (0.777 - 0.629i)T \)
71 \( 1 + (-0.0261 + 0.999i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.958 + 0.284i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−17.85008722598738610589152355520, −17.53591286396136642637483528525, −16.95630867263129560431194346031, −15.99236217706065674163633951583, −15.44700166790650894541241673523, −14.38401182927770670613225006099, −13.67439819404386605041990449020, −13.30297943991348250555555207869, −12.39160731396593928181182476654, −11.88454864318959978484014535307, −11.25057101461492576879446242554, −10.813853718874399068840661548873, −9.95636562761985439652021423284, −9.00265952721979004812685432545, −8.6123230828222861532485900484, −7.80581622571538387752957316498, −7.226624914974015749142669733129, −6.87727494572696317177216071750, −5.7085612762608145292014799134, −4.89749566403552048524545459906, −4.113694455444504081092443111483, −3.07274320951192944947149940132, −2.361286334092940932826919527566, −1.63200990820910994719147890707, −1.21505011485346264779529836944, 0.12969999707168705392668948921, 1.04859365827423054151483509476, 2.33354525879000470965149559206, 2.81176823152412185551877487594, 4.02775693978690190990562334703, 4.84369750660821471632201680011, 5.35824896129735406866876552598, 5.85598064582888615975659631879, 6.795400385422178300672030739228, 7.780328698439654985904964424889, 8.30572144003200380655970563025, 8.657131468981443749298532330061, 9.61691621539922904049333745643, 10.179443199427183397264748026298, 10.78344684281273981995395072528, 11.30655971622134831128052079021, 11.90824083318466192249413265581, 13.21435704562518942850651540742, 13.83097265549914125779881788273, 14.57764652744165638664393808589, 15.13424090271252112324341284666, 15.70162493891684725422715519544, 16.0796940634839192226346557226, 16.9129640298210332250768239375, 17.522558074022158134552175424786

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