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Properties

Label 1-1100-1100.271-r0-0-0
Degree $1$
Conductor $1100$
Sign $-0.991 + 0.131i$
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 + (0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.309 − 0.951i)21-s + (0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.809 + 0.587i)29-s − 31-s + (0.309 − 0.951i)37-s + (−0.809 + 0.587i)39-s − 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 + (0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.309 − 0.951i)21-s + (0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.809 + 0.587i)29-s − 31-s + (0.309 − 0.951i)37-s + (−0.809 + 0.587i)39-s − 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.991 + 0.131i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ -0.991 + 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05363555131 + 0.8092329199i\)
\(L(\frac12)\) \(\approx\) \(0.05363555131 + 0.8092329199i\)
\(L(1)\) \(\approx\) \(0.6984539323 + 0.4240413137i\)
\(L(1)\) \(\approx\) \(0.6984539323 + 0.4240413137i\)

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 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 - T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.844511167278097103768889485361, −20.19594684999543404013723434562, −19.323910062439007611800131799707, −18.78872751257607093212368260986, −18.00121942283342333962850814956, −17.139073797840982125871656407333, −16.55986039566192286423147211023, −15.71308632597270632376050517773, −14.653212612837500595635551152278, −13.72538604659794148209130208751, −13.11654131801729997710768192810, −12.56698821798170940191287398255, −11.57115607644506379777537705533, −10.780610133364602003675206734272, −10.01324510249330078472173558562, −8.85951910377029659900147962605, −8.07151142825408833045461228598, −7.12535774629774192081818396312, −6.526975862858997490580041868991, −5.71736735320361449718702230911, −4.68383903911595098991774960177, −3.40184138178450436806008099830, −2.66057823683431290156153540696, −1.326716973340217156711568496472, −0.37711865917180481245281591180, 1.43580041220627374144143927410, 2.87099181522592420612512094174, 3.62221342265993152900028780904, 4.43208207652154872346033501280, 5.66554217211960319029450866602, 6.03149478155990455324201762126, 7.104638311040930531704054673189, 8.48100992783386993458376039214, 9.032702474442399529682620063925, 9.8500105160739756307189922305, 10.63670942592950593394411116911, 11.37122835085883269138424611624, 12.33279145923217322727590882242, 12.95722040383676593154084572079, 14.18960928512323905321509113188, 14.82715800533143657474743693245, 15.70712355855663611961062019654, 16.27235539679800234967532308965, 16.90780150166460045884160350843, 17.81578100270967392920253422187, 18.83171154967358013662326942763, 19.35797413196960009819846919883, 20.40050321486528018807021134793, 21.23561210440456945080473264391, 21.63682834254196292176638583631

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