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Properties

Label 1-5520-5520.533-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.993 - 0.114i$
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  + (−0.281 − 0.959i)7-s + (−0.755 + 0.654i)11-s + (−0.959 − 0.281i)13-s + (−0.989 − 0.142i)17-s + (0.989 − 0.142i)19-s + (0.989 + 0.142i)29-s + (0.841 + 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (−0.841 + 0.540i)43-s + i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)7-s + (−0.755 + 0.654i)11-s + (−0.959 − 0.281i)13-s + (−0.989 − 0.142i)17-s + (0.989 − 0.142i)19-s + (0.989 + 0.142i)29-s + (0.841 + 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (−0.841 + 0.540i)43-s + i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.993 - 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210886528 - 0.06942024288i\)
\(L(\frac12)\) \(\approx\) \(1.210886528 - 0.06942024288i\)
\(L(1)\) \(\approx\) \(0.9026939508 - 0.06302331411i\)
\(L(1)\) \(\approx\) \(0.9026939508 - 0.06302331411i\)

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

−17.85733313255893782734245249388, −17.350859315688865889214989869243, −16.37370398212953170860330805660, −15.9407335823707861033146438775, −15.26470982056315231387652125233, −14.74169389395295694922410577131, −13.73307453996990477861097385850, −13.37616632009845732957375498913, −12.51576638976310643449772255939, −11.810622542789216294103322850611, −11.46942646919211114929774318205, −10.36322260720148971229265135438, −9.92315349793627026324948657576, −9.06991531541278917116860604513, −8.50710182294061969760059109457, −7.82357460549815361855858849914, −6.95632243535666273641111776242, −6.27946335482577061411643704001, −5.50386448435183232995199839095, −4.953975181263627426956578026302, −4.11952295225760796990181847032, −2.98730814608492694021003384814, −2.63932987925498612290999689138, −1.78315339480322288063810866635, −0.51675904630494739571952048155, 0.574504724184952225384546455162, 1.54893555432875592560755680493, 2.6415217209209356232322178891, 3.0516458308806878422648255752, 4.22044904167701419323829093114, 4.732734358697210244192169409836, 5.3547881217684121162630701987, 6.5382817265738629016862680076, 6.92347178155665503378774394232, 7.770460266032442802256539472082, 8.15473655059246516602082215276, 9.459131447281060206093354417210, 9.7133958540949395441829233729, 10.5025951364542279831328210939, 11.08513943884149280413536634374, 11.90217269535910001919977697318, 12.74596707814655449701558749280, 13.08151528822017037928279338107, 13.99462273227942000360286964156, 14.39276125403142682783141385407, 15.37300597293200742897005326222, 15.87370692177255830385056910385, 16.4509895070782550426461119398, 17.49706334230224462129087136153, 17.61242590747992759598349970154

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