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Properties

Label 1-6025-6025.33-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.975 + 0.217i$
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 + (−0.760 − 0.649i)7-s + (0.891 + 0.453i)8-s + (0.951 + 0.309i)9-s + (0.972 − 0.233i)11-s + (0.891 + 0.453i)12-s + (−0.760 + 0.649i)13-s + (−0.649 − 0.760i)14-s + (0.809 + 0.587i)16-s + (−0.382 − 0.923i)17-s + (0.891 + 0.453i)18-s + (0.996 + 0.0784i)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 + (−0.760 − 0.649i)7-s + (0.891 + 0.453i)8-s + (0.951 + 0.309i)9-s + (0.972 − 0.233i)11-s + (0.891 + 0.453i)12-s + (−0.760 + 0.649i)13-s + (−0.649 − 0.760i)14-s + (0.809 + 0.587i)16-s + (−0.382 − 0.923i)17-s + (0.891 + 0.453i)18-s + (0.996 + 0.0784i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.975 + 0.217i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.975 + 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.764881724 + 0.6356384205i\)
\(L(\frac12)\) \(\approx\) \(5.764881724 + 0.6356384205i\)
\(L(1)\) \(\approx\) \(2.786931574 + 0.2804178160i\)
\(L(1)\) \(\approx\) \(2.786931574 + 0.2804178160i\)

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

−17.646483357237461635012416298267, −16.959684427551015401476522970855, −15.94505813238327420883622313641, −15.54740684071687101623721547222, −14.962261596558518831894275338484, −14.4310949870469256295658475116, −13.74519728202528919940246201347, −13.10247306283112296817054736091, −12.49495453606192466612737566936, −12.124429041114476332309339162930, −11.25760329728188875138255903522, −10.345643996267339815362630165093, −9.5899508443924748888119353231, −9.24527175457179820486582279447, −8.253080779480584340504651235032, −7.36520449612499376123737949612, −7.007063080008157925583287375463, −6.03948444872418466285908032596, −5.56528867163376658709315073480, −4.49378403800090258033908409466, −3.86702982454734678933167216415, −3.19196942159198207569167754583, −2.608156438023930196198251931388, −1.88348158598663678842303547414, −1.03742809918627227352125531653, 0.98120669545812867452272811101, 1.9158288344345898910059208120, 2.77801642596221872128812130316, 3.3112179836146656550557827045, 3.965294057394807147142485300224, 4.635934805829852815921536490189, 5.27167130086039297335179541919, 6.53084235680266568385781556119, 6.91208176526892544589182494613, 7.35438212655871146949783742887, 8.31002217769232370468511298277, 9.17730583995830423253984798446, 9.6378057746877896097273994141, 10.485056616373838962504707038156, 11.18749214118869419982728734283, 12.10952350383664753208747645133, 12.52973651585987746146717581004, 13.37579473295136172394497564409, 13.965876937537200939714401933757, 14.27660561316906861857186199459, 14.86013371455308364732846624473, 15.846461922174441235734589744289, 16.175650615744467848883777735754, 16.74718608763022559826432318997, 17.53828379940034268599059892668

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