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Properties

Label 1-1100-1100.459-r0-0-0
Degree $1$
Conductor $1100$
Sign $-0.0561 - 0.998i$
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  + (0.309 − 0.951i)3-s + (0.809 + 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + 19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)27-s − 29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (0.809 + 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + 19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)27-s − 29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0561 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0561 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.0561 - 0.998i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ -0.0561 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150297014 - 1.216745708i\)
\(L(\frac12)\) \(\approx\) \(1.150297014 - 1.216745708i\)
\(L(1)\) \(\approx\) \(1.132382094 - 0.4692581809i\)
\(L(1)\) \(\approx\) \(1.132382094 - 0.4692581809i\)

Euler product

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

−21.58546649679513201066468947772, −20.798871752924775021108712127600, −20.23137671329527850417470955787, −19.48488717683766513098027193465, −18.56919080289037292558464478245, −17.50865918745792458126337964917, −16.83711058103708446757594224438, −16.31596298298589680870773472366, −15.16524976305838961000356392021, −14.64020975996269364618907243536, −14.03804634114568506514546279676, −13.11378103854664919646719225200, −11.90210642457505018408741174104, −11.25741335780340877164060905311, −10.35100031634060847349330813952, −9.80123377900735487158090925289, −8.80699240589857068094624673725, −8.037811848681195239469738880634, −7.24304594244279815664704509652, −6.02920222491310085562298078934, −4.927645428679656288339038241890, −4.45680059384082345849015537638, −3.47825726676750772330782244036, −2.47367786835004082091089970968, −1.28820921605670868827988995110, 0.70891242410472822688023447438, 1.8724560306882041983904773729, 2.64484439747864748622590081884, 3.59530690893811317490016393448, 5.1816357247016517585619237187, 5.49335441176364585479616312504, 6.813543792350400664353190151262, 7.606641966005336621623478093006, 8.09668025030888224472774684431, 9.17404652489086981184577415611, 9.82654968512597864596162211853, 11.2876067538676441851140072849, 11.73020679330827421393828398035, 12.49842011590756211983589491618, 13.40621575101627553672151116444, 14.09891883672328445141491794415, 14.886766126886539629942880353156, 15.518151766378341584265936952890, 16.74856132961733065138346931323, 17.559865014819137636752275685533, 18.16049582042837503441307074042, 18.77597425750405115779190557379, 19.63697036326416423934394490070, 20.40237039456351206668993794677, 21.0016103138272626280143748933

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