↑

Properties

Label 1-6025-6025.1504-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.559 - 0.828i$
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  + (−0.913 + 0.406i)2-s + (−0.913 − 0.406i)3-s + (0.669 − 0.743i)4-s + 6-s + (0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.913 − 0.406i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (−0.913 − 0.406i)3-s + (0.669 − 0.743i)4-s + 6-s + (0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.913 − 0.406i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1504, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4704868687 - 0.2501014498i\)
\(L(\frac12)\) \(\approx\) \(0.4704868687 - 0.2501014498i\)
\(L(1)\) \(\approx\) \(0.5186966759 + 0.008087057621i\)
\(L(1)\) \(\approx\) \(0.5186966759 + 0.008087057621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
7 \( 1 + (0.913 - 0.406i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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

−17.678399048427726840974778264439, −17.41738945700645634754498146785, −16.67306544382123913882995167640, −15.89208208581609463651764407277, −15.41260362314861420606238711389, −14.978277997765056875861186107270, −13.685900916115811264948098539864, −13.00464448059548439097809140375, −12.17864011771880783898725494285, −11.81080705053649320719298479500, −10.95886564405515985169124066340, −10.626769854542186112772832087234, −10.09245089987614332594751512753, −9.1351937343980141715057403079, −8.59975696025692577848679518590, −7.77442426275139732223234719298, −7.304007191193869001464136678455, −6.25709573447200673046798244409, −5.6328224553280380633405787748, −4.946611554871869817511934243549, −4.1384951111985791595814607396, −3.19006208789488137478479984315, −2.43673788521099512758945872641, −1.55001567463191704807684363046, −0.65769201365064589647366354111, 0.316219887128502728355018673678, 1.34254712205106016537358643436, 1.991864493476043039976283547392, 2.632333589177634123456500204135, 4.4485951279961816185332407198, 4.64331042660202476977884674165, 5.49263210589818825311222896238, 6.38606431189093386065477061695, 6.91591219580904467710458304368, 7.36676044927380609413793243668, 8.37509244228969232601303424530, 8.59706037833810471593011996007, 9.76608834274527259135922027975, 10.449793147102756165616776479165, 10.9457197367752391494415134005, 11.35444373529846722746932246927, 12.23267577639034947826335963332, 12.87280009496386294811911920574, 13.79564472950843839426950349412, 14.390737983701722891611032589898, 15.24706825473508352034827511466, 15.80881304357568856943488432169, 16.46056051108285099327001134428, 17.15700965070950479414702364243, 17.66621388739421401311447328307

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