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Properties

Label 1-4560-4560.539-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.0818 - 0.996i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  − 7-s − i·11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)29-s − 31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + 49-s + (0.866 − 0.5i)53-s + (−0.866 − 0.5i)59-s + (−0.866 + 0.5i)61-s + ⋯
L(s)  = 1  − 7-s − i·11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)29-s − 31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + 49-s + (0.866 − 0.5i)53-s + (−0.866 − 0.5i)59-s + (−0.866 + 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0818 - 0.996i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.0818 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5157367315 - 0.5598348648i\)
\(L(\frac12)\) \(\approx\) \(0.5157367315 - 0.5598348648i\)
\(L(1)\) \(\approx\) \(0.8506072247 + 0.01828082032i\)
\(L(1)\) \(\approx\) \(0.8506072247 + 0.01828082032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - T \)
11 \( 1 - iT \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−18.46211204977487789301500786314, −17.911722802563336388426315373676, −16.73401909987462464178013238660, −16.50535460107562282804274843691, −15.726822415544270102251945122945, −15.30872123197032239042672838438, −14.12482084644940498826958986840, −13.584750970583933984583270932877, −13.29507349066105754070472743950, −12.19863009524073712419391074908, −11.69937122862297858919957603930, −10.89418194017609078065710067902, −10.291906952967977795678149939756, −9.36037308858618156794977695116, −8.931607898570367474791945768609, −8.20238908256318142163532166122, −7.20115165278260779197721919812, −6.599254051829815539168893262526, −5.9488707070922756968739939039, −5.241656206958753444110713168986, −4.217855611800197304783062859529, −3.4330208016014188273309515137, −2.96921807068149832746197167422, −1.86887368487830122718752587601, −0.89428566059167906690491046004, 0.24019702853133538261195823180, 1.48068992424410941673295252197, 2.31094051985240485923511489543, 3.16506778500271000129278096307, 3.9573050150516660253624823949, 4.55706135598969834360876108854, 5.657986705787858096796994561407, 6.257074932121991772733673440286, 6.85870137909247921019886208589, 7.68819950823268233414125533074, 8.51079560139815775854871753279, 9.11512557658345613648874297755, 9.99644886804557697622177150544, 10.42255016968163204320845421915, 11.15989937773645543337384549329, 12.21016971411386441337382083316, 12.61985837380528261332251864351, 13.24712124518341595332899202525, 13.908459807819838423280532728161, 14.89432545020629891485408022056, 15.33111913397961838267933122416, 16.06459660951654608160370002824, 16.61588470757254346751313371769, 17.440743203562016539775338151651, 18.08811173363961791174916521048

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