Properties

Label 1-6025-6025.28-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.751 + 0.659i$
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 + (−0.707 + 0.707i)3-s + (0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (−0.522 + 0.852i)7-s + (0.891 − 0.453i)8-s i·9-s + (−0.233 + 0.972i)11-s + (−0.453 + 0.891i)12-s + (−0.0784 + 0.996i)13-s + (−0.382 + 0.923i)14-s + (0.809 − 0.587i)16-s + (0.0784 − 0.996i)17-s + (−0.156 − 0.987i)18-s + (0.0784 + 0.996i)19-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.707 + 0.707i)3-s + (0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (−0.522 + 0.852i)7-s + (0.891 − 0.453i)8-s i·9-s + (−0.233 + 0.972i)11-s + (−0.453 + 0.891i)12-s + (−0.0784 + 0.996i)13-s + (−0.382 + 0.923i)14-s + (0.809 − 0.587i)16-s + (0.0784 − 0.996i)17-s + (−0.156 − 0.987i)18-s + (0.0784 + 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7636363108 + 2.028481043i\)
\(L(\frac12)\) \(\approx\) \(0.7636363108 + 2.028481043i\)
\(L(1)\) \(\approx\) \(1.330757651 + 0.5655290321i\)
\(L(1)\) \(\approx\) \(1.330757651 + 0.5655290321i\)

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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.522 + 0.852i)T \)
11 \( 1 + (-0.233 + 0.972i)T \)
13 \( 1 + (-0.0784 + 0.996i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (0.0784 + 0.996i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (-0.996 - 0.0784i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (0.891 - 0.453i)T \)
43 \( 1 + (0.852 - 0.522i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.760 - 0.649i)T \)
73 \( 1 + (-0.0784 - 0.996i)T \)
79 \( 1 + (-0.987 - 0.156i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.852 + 0.522i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.29345819790061783575811434365, −16.79115507377024523830095002917, −16.03379228028792782540888425710, −15.72883679401479348752148682810, −14.60673856658976980840699221743, −14.08293007664743697772649033939, −13.30202582927710102138892646875, −12.89489437974281732490547276474, −12.54807410512837945425829513329, −11.533821437515276593161175446187, −10.87401058220034264884132621307, −10.65359607186362005813569699260, −9.70714787964659201799096248183, −8.31844612452568394881374775899, −7.90147859413730687466830212329, −7.19219383756391589323206654386, −6.43685150286995201451163675257, −5.981204906968024837675501049960, −5.40131684144501570608741707860, −4.469280160755381104774874727160, −3.861005033048461323287328225886, −2.9002404528637440287758388422, −2.358258641931670217350175352752, −1.15360820290318202091123929017, −0.43833189116335588621400627350, 1.20463439279186534206902557429, 2.1629550148876901397508653117, 2.838500517524419311688999600, 3.800186337824244442119507725837, 4.26375302560618629197627981019, 5.09529065782920924319509457788, 5.64872405684178950268251298452, 6.16246636971382016023326719270, 7.07206877077352356147632428206, 7.50873664628528449658459658291, 8.98514445602275104597271025140, 9.44795653186516559440525068753, 10.13058286269534491237449382615, 10.76023252239692594881531834942, 11.66018977981946086079229856661, 12.05383744332017475447342832495, 12.44454812983092399055609127151, 13.29060887307044408406975506861, 14.1792441843432030941532781029, 14.721334735185141214946461079070, 15.35194229783409111420406229, 16.037309582557270866589912594666, 16.30119283991752082649499195608, 17.03998759222493840459373873729, 18.08756853144512719167249150557

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