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Properties

Label 1-5520-5520.3899-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.419 - 0.907i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (0.142 − 0.989i)7-s + (−0.909 + 0.415i)11-s + (0.989 − 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (0.841 − 0.540i)41-s + (0.281 − 0.959i)43-s − 47-s + (−0.959 − 0.281i)49-s + (−0.989 − 0.142i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)7-s + (−0.909 + 0.415i)11-s + (0.989 − 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (0.841 − 0.540i)41-s + (0.281 − 0.959i)43-s − 47-s + (−0.959 − 0.281i)49-s + (−0.989 − 0.142i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6823836885 - 1.067415319i\)
\(L(\frac12)\) \(\approx\) \(0.6823836885 - 1.067415319i\)
\(L(1)\) \(\approx\) \(0.9808064863 - 0.2166244975i\)
\(L(1)\) \(\approx\) \(0.9808064863 - 0.2166244975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.909 + 0.415i)T \)
13 \( 1 + (0.989 - 0.142i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.755 - 0.654i)T \)
29 \( 1 + (-0.755 - 0.654i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.540 - 0.841i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.989 - 0.142i)T \)
59 \( 1 + (0.989 - 0.142i)T \)
61 \( 1 + (0.281 + 0.959i)T \)
67 \( 1 + (-0.909 - 0.415i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−18.069244922985142690845741989, −17.74104284664743728977195963177, −16.48749507692721920898754658645, −16.045208637233765996468686032280, −15.6296182810847818545466789741, −14.78314365655265744079298758976, −14.151233920936892893053012703539, −13.316695891815341262822311733586, −12.937828829011949274295147639314, −12.00910681950412358943754238275, −11.367245106485086062189248858882, −10.95751780209876112320226710210, −9.93692259138849665881647056733, −9.366664606642192305169193434585, −8.55464921712894949892193883991, −8.10906875635446056858442483731, −7.31655120451567296274765183546, −6.29712992777692797441591976071, −5.86423627367582439390001364618, −5.058796179153495516198056890718, −4.44990984840291157096024159027, −3.22199164990717204932061590233, −2.89530230405444157250066160323, −1.89272276461517825002106550834, −1.064348654266572383071474917977, 0.351900508371459159320424024772, 1.28162946835247070302708231408, 2.183718289688479197219834856197, 3.021833027039040850858063409696, 3.971446669240287497669791850016, 4.382819835101097478661036564996, 5.347601551247813682747085179430, 6.0244478179153480961244910198, 6.87938960279848444303982543422, 7.51656659024317619199190407312, 8.118581289766908607990177743, 8.85389472655229353006290767225, 9.75251332491018698380334890045, 10.316417409666661995546772274789, 11.0783155445262242897993579087, 11.37461699299622050871408668071, 12.53565662790987337133629738147, 13.17335560026716263249028405454, 13.5420790239467556533040901262, 14.27161323785506201221950774888, 15.15927936923799745531911601599, 15.67718463703817547249685180329, 16.28000530535164841380262882269, 17.06464183223233569665839543449, 17.858721966316784425308666124839

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