↑

Properties

Label 1-6025-6025.64-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.806 - 0.591i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)12-s + (0.951 − 0.309i)13-s + (0.951 + 0.309i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + 18-s + (0.587 + 0.809i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)12-s + (0.951 − 0.309i)13-s + (0.951 + 0.309i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + 18-s + (0.587 + 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155746297 - 0.3784897783i\)
\(L(\frac12)\) \(\approx\) \(1.155746297 - 0.3784897783i\)
\(L(1)\) \(\approx\) \(0.7738215251 - 0.4113335542i\)
\(L(1)\) \(\approx\) \(0.7738215251 - 0.4113335542i\)

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 \( 1 + iT \)
11 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.64727443091337232072711082922, −16.99224284618768884955530192897, −16.22943294885169823855049880638, −16.04213405816933464268096061287, −15.36890474037444233990135249799, −14.59306277746564524935055334026, −13.85595109451281145670140876306, −13.332740299595816655553167271040, −12.64721559774350417235634413428, −11.89731708931688665628217784935, −11.039160014636864257755079175636, −10.584543670279693339397294743112, −9.65662016792273547044810127237, −9.252386846297328544693633640669, −8.238726986870820934628920741488, −7.53490110283741752341899846432, −6.85824893485809305261439863299, −6.37460663401291232854249293540, −5.46951918215438526537389367837, −4.72710080612697172915728067661, −4.594513546660971104443402205840, −3.387376393199903722447104275127, −3.06677305730952119988703864675, −1.27045562124308321688165522265, −0.47254699698113973344047676584, 0.734676547515105830877545558440, 1.604262066455714118168805902017, 2.15925943768875130366326635436, 3.19746104322602361145345472095, 3.64615422023991676716554622608, 5.02367829421424221467187846420, 5.30357018143339425508987530121, 5.89851338428863887393485402715, 6.56073110269673917412645031576, 7.75902659607966558885122670175, 8.324256687807734181056996995797, 8.98574499767863718384185522232, 9.98053496149222353229137808176, 10.614728684291354959466751372298, 11.02082381099098105959967295741, 11.88978709392304225225992671326, 12.2587007665915929302543260023, 12.9425625931078550905988813345, 13.39765561930260783391402513562, 14.120547718612844658342518113748, 15.02023938057028242429158144398, 15.65140939511424726466603560730, 16.333650676448867297782144528720, 17.21517397190231327572245449691, 17.93851275257650554575641017947

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