Properties

Label 1-6025-6025.3317-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.996 - 0.0869i$
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 + (0.987 + 0.156i)3-s + (0.951 + 0.309i)4-s + (0.951 + 0.309i)6-s + (0.649 − 0.760i)7-s + (0.891 + 0.453i)8-s + (0.951 + 0.309i)9-s + (−0.233 − 0.972i)11-s + (0.891 + 0.453i)12-s + (0.649 + 0.760i)13-s + (0.760 − 0.649i)14-s + (0.809 + 0.587i)16-s + (−0.923 + 0.382i)17-s + (0.891 + 0.453i)18-s + (0.0784 − 0.996i)19-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.987 + 0.156i)3-s + (0.951 + 0.309i)4-s + (0.951 + 0.309i)6-s + (0.649 − 0.760i)7-s + (0.891 + 0.453i)8-s + (0.951 + 0.309i)9-s + (−0.233 − 0.972i)11-s + (0.891 + 0.453i)12-s + (0.649 + 0.760i)13-s + (0.760 − 0.649i)14-s + (0.809 + 0.587i)16-s + (−0.923 + 0.382i)17-s + (0.891 + 0.453i)18-s + (0.0784 − 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.996 - 0.0869i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.996 - 0.0869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.574541125 - 0.2864213368i\)
\(L(\frac12)\) \(\approx\) \(6.574541125 - 0.2864213368i\)
\(L(1)\) \(\approx\) \(3.083476677 + 0.1087396417i\)
\(L(1)\) \(\approx\) \(3.083476677 + 0.1087396417i\)

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 \( 1 + (0.987 + 0.156i)T \)
7 \( 1 + (0.649 - 0.760i)T \)
11 \( 1 + (-0.233 - 0.972i)T \)
13 \( 1 + (0.649 + 0.760i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.0784 - 0.996i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.156 + 0.987i)T \)
43 \( 1 + (0.760 - 0.649i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.649 - 0.760i)T \)
79 \( 1 + (0.891 + 0.453i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.852 - 0.522i)T \)
97 \( 1 + T \)
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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

−17.891209351879365490638705631094, −17.04919593897782240724441372270, −15.79919026710019845203628681177, −15.64893014561241804349799047937, −15.01568507441939454789847134586, −14.36351691677117587910249645777, −13.94858929443078287968419511918, −13.05719617439279465620780565961, −12.474397733081159612383597553, −12.26088177969135017365220592030, −10.94484973360993801924291880401, −10.774164210639266420195029179972, −9.70283189697038816880598914236, −9.026555280467690007693199088513, −8.26769491002039234539063568000, −7.59601815019013640492877496535, −6.96329567872005873245602580333, −6.19835357843688448500464275802, −5.21916216637692969198696552768, −4.82563910024265672047257875438, −3.90953785841226669839764027796, −3.234288628993781645950938969264, −2.51347078585098415004387892735, −1.863121517457278483217104037878, −1.261463195496491988617900795053, 0.99246349080236512852057228389, 1.913837947967049030051153876230, 2.50334830054965213612622826919, 3.47975244784530074543487252671, 3.93500579374489302745884209640, 4.55271622193751619922840667186, 5.307156814345356698336493845, 6.212738449963319340127367730281, 7.06468855626817665410589380874, 7.42178550476048577229025285505, 8.39317460614793944257764959186, 8.75727936363730997461427960963, 9.69888374628011734912845287208, 10.782767486899509414635819890171, 11.07301499535738956950165570307, 11.64040930308596938723235074692, 12.89105976750477928178290676981, 13.35184712787191866250107843969, 13.71794119010907413551232963931, 14.2824544212872187301798170796, 15.06905465551824250906618494123, 15.496235059213601717012739602514, 16.18561872047032818153457053170, 16.87727986369407820340116465990, 17.49076159976217341352219487630

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