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Properties

Label 1-6025-6025.781-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.975 - 0.221i$
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  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.913 + 0.406i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.104 + 0.994i)11-s + (0.669 + 0.743i)12-s + (0.978 − 0.207i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)16-s + (−0.309 + 0.951i)17-s + (0.913 + 0.406i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.913 + 0.406i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.104 + 0.994i)11-s + (0.669 + 0.743i)12-s + (0.978 − 0.207i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)16-s + (−0.309 + 0.951i)17-s + (0.913 + 0.406i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9947643095 - 0.1114791189i\)
\(L(\frac12)\) \(\approx\) \(0.9947643095 - 0.1114791189i\)
\(L(1)\) \(\approx\) \(0.6707896620 - 0.3263920714i\)
\(L(1)\) \(\approx\) \(0.6707896620 - 0.3263920714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−17.52584144788643786957858744691, −16.66174215683189904555874537128, −16.505308927324573036650643530222, −15.78597358214811388625639185918, −15.56394236553805219465681367823, −14.43860508041204288516924929689, −13.99587154103835315732594314958, −13.3237558734205497382465587977, −12.65245483805655232871317508739, −11.675928646746159868619174458463, −10.87034965767032372673254664060, −10.49509774211304868096939266357, −9.469173754129346388971670160569, −9.06773027989630071519007356104, −8.67252719766882687975079732013, −7.3642935361535731778920709610, −6.98662848537733831153435684671, −6.06472858614746546302867462651, −5.69385210869373992307133200456, −4.97041715560021881648423852223, −4.0369612407432561273642582373, −3.58292817329213903307973770025, −2.86020805660959193654901891034, −1.03653707522146507068274265310, −0.44910646348785841368839913544, 0.806457538997551630658202325285, 1.59900225267027023941478414201, 2.20648307275637062846736313703, 3.12192368243716132481592595339, 3.73799459167357667671525521991, 4.67827964485942799264134449689, 5.58956144484376107666799941836, 6.0215594683624329407099116930, 6.95485319705944557645595805024, 7.67467494518441537068965812110, 8.37147814706289600935667004200, 9.23739592343074951900633489732, 9.70514387902912248061545204319, 10.5450639978529093973778981599, 11.25393977968006603284765801415, 11.6899689083238766957493718009, 12.599605237364865692518218184438, 12.874578259392351840196061751157, 13.350248002422968630907220608084, 14.12490113455732576987367107775, 15.04753561971825327226520918990, 15.72923466194955576627632851216, 16.73540831363603814346460833760, 17.1414011062664729748006483444, 17.97534929788281604259340837300

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