Properties

Label 1-5520-5520.203-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.879 + 0.475i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262285428 + 0.3189591967i\)
\(L(\frac12)\) \(\approx\) \(1.262285428 + 0.3189591967i\)
\(L(1)\) \(\approx\) \(0.9521244942 + 0.02462418995i\)
\(L(1)\) \(\approx\) \(0.9521244942 + 0.02462418995i\)

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

−17.88682795664156905857758662455, −17.18783369034640014071159130383, −16.227355654478548986737473005533, −16.02121636237958697238121136459, −15.25742343650927491323374318147, −14.389995755103133374170375617242, −14.03440178373205995710957945631, −13.11704694967106734951416332418, −12.270790030392966470405413987111, −12.11243613553855281769599004594, −11.18447913943627060416083360, −10.40940501687055730220501436762, −9.72384939763511476192320981706, −9.13891557802102133895419275718, −8.383253133318851913747673783134, −7.67514092122045858016038721255, −7.03129049346343664274008015999, −6.10209524888011995435694722674, −5.300398159856290423514378141730, −5.14298839288478398149954221138, −3.851575454739553670904152491515, −3.06021924877903457381263304224, −2.54698040401100687416911521438, −1.62509770980322572637264242934, −0.45293521826372507228346844150, 0.73504761922353043473511107015, 1.596928680142644377881789645169, 2.68473704232315531019474739750, 3.23558920969958362813876405179, 4.13215085633492995812399008823, 4.952618990012075313756684912241, 5.40826750299835398889633864480, 6.47102353709952008974874679728, 7.22132876778105587713787151499, 7.65136264556987025407970442346, 8.29395666887312973587824234890, 9.46119882307043292107695239394, 9.988635090196830979042502796877, 10.339252246599652706973897914319, 11.22724214158833742123112223350, 12.11074846905158934797792449374, 12.52289558468187624058821941223, 13.45724850247813518305700743314, 13.75719971865096544344470617846, 14.84640912748866504078859492583, 15.04389490261395196044198953814, 16.15896520126196250873246504944, 16.54157208199063242689012981747, 17.226079231133576073444862202661, 17.951127050623551192052950982900

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