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Properties

Label 1-220-220.87-r1-0-0
Degree $1$
Conductor $220$
Sign $-0.850 - 0.525i$
Analytic cond. $23.6422$
Root an. cond. $23.6422$
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  + i·3-s + i·7-s − 9-s + i·13-s − i·17-s − 19-s − 21-s + i·23-s − i·27-s + 29-s − 31-s + i·37-s − 39-s − 41-s − i·43-s + ⋯
L(s)  = 1  + i·3-s + i·7-s − 9-s + i·13-s − i·17-s − 19-s − 21-s + i·23-s − i·27-s + 29-s − 31-s + i·37-s − 39-s − 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(23.6422\)
Root analytic conductor: \(23.6422\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 220,\ (1:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1849995754 + 0.6512257044i\)
\(L(\frac12)\) \(\approx\) \(-0.1849995754 + 0.6512257044i\)
\(L(1)\) \(\approx\) \(0.7206922346 + 0.4454122964i\)
\(L(1)\) \(\approx\) \(0.7206922346 + 0.4454122964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + iT \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−25.64197351126417876224403432024, −24.80769965850903725230043872004, −23.74487010530312504915880778103, −23.23969794445472054328918622829, −22.21615745758608820099183436369, −20.85044292659556319552620332410, −19.90853029912981745610179643094, −19.27621806996160300227395151265, −18.0714057200208878316630707065, −17.3363858391224685791395466047, −16.486674235188267383485232586801, −14.97109699417453614694131297102, −14.1208603985950621568014783228, −13.00351250745034291532488145400, −12.51364981327962613062609684879, −11.02728002115044801874492667965, −10.31392297805618772966197588514, −8.637340942124016993401018642283, −7.81821079114914794495333309741, −6.78785425770845287785846729243, −5.83190642115220243582811012746, −4.28957187804958529223648028723, −2.90531553818759513292302283825, −1.47685027556573616472849437375, −0.21849873347784152543807740305, 2.10542880675252616041689561865, 3.35560900886508211492350056127, 4.63159435568625500058012748923, 5.547485906144781637147377431222, 6.782699272053790793186550432486, 8.44847237498877655564291049589, 9.17136860299736831197040811751, 10.11166094507863638474789841808, 11.3876765260545879802514168802, 12.01742007267299684545257888769, 13.542141570274725739331113960, 14.59977952069673774116411298516, 15.43917133943318067469876643182, 16.245724883291026392195010328124, 17.19806292716654172428256985, 18.384464951924386260818398998015, 19.34612683107131300177709672972, 20.44390731998101943116711243126, 21.47217580777660006707620472857, 21.86459050947982757330527255715, 22.99355058797063046268282615366, 23.971101563544255541531191505975, 25.327841853039922784304440444807, 25.76970126310737634423835027747, 27.07458884282656845986739421508

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