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Properties

Label 1-1100-1100.367-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.996 + 0.0864i$
Analytic cond. $5.10837$
Root an. cond. $5.10837$
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  + (−0.587 − 0.809i)3-s + (0.587 + 0.809i)7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)13-s + i·17-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.587 − 0.809i)23-s + (0.951 − 0.309i)27-s + (−0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + i·37-s + (0.309 + 0.951i)39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + (0.587 + 0.809i)7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)13-s + i·17-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.587 − 0.809i)23-s + (0.951 − 0.309i)27-s + (−0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + i·37-s + (0.309 + 0.951i)39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.996 + 0.0864i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.996 + 0.0864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.127137967 + 0.04883885865i\)
\(L(\frac12)\) \(\approx\) \(1.127137967 + 0.04883885865i\)
\(L(1)\) \(\approx\) \(0.8972877620 - 0.08650627619i\)
\(L(1)\) \(\approx\) \(0.8972877620 - 0.08650627619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 - iT \)
59 \( 1 + T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + iT \)
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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

−21.48770136525636921043213445039, −20.55428384379745815055633728336, −20.19448242837105030247749314664, −19.09603956570071478504431712039, −18.022609157193626294294851514910, −17.4629958211613865655482133091, −16.70165143176736986471632731199, −16.09322385610924985764127531835, −15.2365947080284628089282508128, −14.23542530113124757767371555213, −13.92854004173343719249535340891, −12.44062889406083587557834141506, −11.85144114862847451251741380000, −11.04127145278918512330574588378, −10.249933973464029041442686675294, −9.64724653995564235832353924278, −8.70922208768751426903354628341, −7.51309584476942007169280020147, −6.93574200253059266179100012739, −5.627814486560248918737722566241, −5.035278462910080680255441071807, −4.14257015592113048978610003113, −3.38657356880477199620590285310, −1.99191536928472774078186844091, −0.65951079881669850432063270450, 0.90896833299965556395272534294, 2.13802385518708215054890905002, 2.703913115537806469590074483497, 4.387197810114311876011592473875, 5.13827685267434797585285806046, 6.01523143362820505004350305282, 6.706270562655368561498123308832, 7.90816395477521465838075664127, 8.21126048887023194435995949808, 9.463547274659250044172034410208, 10.40220113082239989433383069204, 11.40222760138884244726771421945, 11.87986449341615002560075722621, 12.71770422729367900451761162033, 13.356644095886484186225910349750, 14.45025442998765043209553086726, 15.07953002938887461044299735075, 16.01695252236173652086319891838, 17.06395398989794132070925746088, 17.552579025967595414878787798292, 18.24130878632732528784323244920, 19.07954062646908855468735151103, 19.62639403101885687820187248444, 20.64308979847029258077606673922, 21.64152485110995977240646590888

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