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Properties

Label 1-5520-5520.1499-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.425 - 0.905i$
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  + (0.959 − 0.281i)7-s + (0.755 − 0.654i)11-s + (0.281 − 0.959i)13-s + (−0.142 + 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (−0.909 − 0.415i)37-s + (0.415 + 0.909i)41-s + (0.540 + 0.841i)43-s − 47-s + (0.841 − 0.540i)49-s + (−0.281 − 0.959i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)7-s + (0.755 − 0.654i)11-s + (0.281 − 0.959i)13-s + (−0.142 + 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (−0.909 − 0.415i)37-s + (0.415 + 0.909i)41-s + (0.540 + 0.841i)43-s − 47-s + (0.841 − 0.540i)49-s + (−0.281 − 0.959i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.425 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.837079515 - 1.166855799i\)
\(L(\frac12)\) \(\approx\) \(1.837079515 - 1.166855799i\)
\(L(1)\) \(\approx\) \(1.256375514 - 0.2275612081i\)
\(L(1)\) \(\approx\) \(1.256375514 - 0.2275612081i\)

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 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (0.755 - 0.654i)T \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (-0.989 - 0.142i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (-0.909 - 0.415i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.281 - 0.959i)T \)
61 \( 1 + (0.540 - 0.841i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−17.957887967939650764511581432902, −17.43409426788625110917880266421, −16.657369677056267541768815111023, −16.04612971709704674837531210570, −15.31837835503301327298964529270, −14.58507465786866750219750661535, −14.08436259978460221288852815999, −13.58727852728478695399170101034, −12.482494465623916983619963050595, −11.93090190312335800469470600857, −11.43471958138919903337800555717, −10.801731833068594350485560069617, −9.83969282069592957071587322775, −9.08686839784079477949239942854, −8.83528252736468727209543613858, −7.67861544078189397740931585410, −7.21445409209865392872232755085, −6.5465611740408176127055530108, −5.45734551581385251224247523646, −5.05413466453672964620469238185, −4.15699178280347642463434216378, −3.57002585941544478041142485473, −2.416138804992017760497720790367, −1.76262627590521626441393433145, −1.05983588637870500915004884469, 0.61062908028514204257177392676, 1.44831805642180827950427779338, 2.139821882522640901416670116, 3.4379971176827486731953466867, 3.67709588878635875605832038371, 4.74315729528753519837585741727, 5.42738350106341027465911120137, 6.070583606452515198216911333173, 6.87800843792914306021527301660, 7.8920767821814642256424602006, 8.059469485669599429744208197474, 9.007763651532568793121185442679, 9.65342753741888186458671013053, 10.54287832627170669136606308456, 11.2487922627602534041275038038, 11.45038438522897315780898355623, 12.55532552422604273079251003131, 13.09140954409727516969280995909, 13.85821299978390235428926697444, 14.53121671138486939683299512698, 14.94999401054038479040163172860, 15.78723026627585282113503859245, 16.50951295136516237799041305455, 17.12895095229486779675246788001, 17.811672726152838589769785785379

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