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Properties

Label 1-6025-6025.604-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.587 + 0.809i$
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 + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.809 − 0.587i)17-s + (0.978 + 0.207i)18-s + (0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.809 − 0.587i)17-s + (0.978 + 0.207i)18-s + (0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1277817846 + 0.06514968571i\)
\(L(\frac12)\) \(\approx\) \(0.1277817846 + 0.06514968571i\)
\(L(1)\) \(\approx\) \(0.4953330774 - 0.3841860765i\)
\(L(1)\) \(\approx\) \(0.4953330774 - 0.3841860765i\)

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 \( 1 + (-0.669 - 0.743i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
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 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
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 \( 1 + T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−17.37315302782751914463905265541, −17.07184163106417625231653814623, −16.05000354222196513968590909478, −15.755306815308681231449394527750, −15.29865009832499499443311121959, −14.48361997001761344638620042808, −13.8981116990469361996740091735, −12.85605524688737580017379750920, −12.49568781573763630743077132782, −11.9546924180652496230619953664, −10.56955440238413401143900490958, −10.31980085274248197925819814835, −9.60503416796848482409190538819, −9.01235508352147568100217362205, −8.26544674200346155437145772230, −7.15376174982020268291933311858, −6.876564217274924255280299100945, −6.099542260986482165182637361460, −5.39452345947956660046803453310, −4.79771392870421586147826802301, −4.08754150014142754198318387363, −3.45056845743930861024299267628, −2.57017674553274297813427742847, −0.99296622302202313735104023791, −0.05618351384045798536987420521, 0.8808442495176144579843108094, 1.628158233741380258397386155136, 2.7038740307742553213309723497, 2.9835505186699187713801609261, 4.08230261598379883342541138769, 4.85319842698655382520412431960, 5.59725266871555680651862997685, 6.209554548351810750728015460398, 6.963819025852489556251973748241, 7.69867507444007953060495404773, 8.70906955697034156930231971925, 9.27251293811771358451815913357, 9.87132179856770619172993801544, 10.77306268869890777746436500949, 11.35963974609875873007479555369, 11.87433490528633026327726209901, 12.36103920447843221747302596959, 13.22711764919471518428824928033, 13.68869064012819700469238642221, 13.99539952834713461066507846265, 15.19207193371908041128161213639, 16.057878097538586431338601663930, 16.588060952555683439133037310101, 17.32428418478127222385332820742, 17.95960827186544841902859488496

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