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Properties

Label 1-1100-1100.647-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.945 - 0.325i$
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  − i·3-s + (0.951 − 0.309i)7-s − 9-s − i·13-s + (−0.951 + 0.309i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s − i·27-s + (0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 − 0.809i)37-s + 39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  − i·3-s + (0.951 − 0.309i)7-s − 9-s − i·13-s + (−0.951 + 0.309i)17-s + (−0.809 − 0.587i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s − i·27-s + (0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 − 0.809i)37-s + 39-s + (0.309 + 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395440054 - 0.2332432690i\)
\(L(\frac12)\) \(\approx\) \(1.395440054 - 0.2332432690i\)
\(L(1)\) \(\approx\) \(1.085414692 + 0.1112184370i\)
\(L(1)\) \(\approx\) \(1.085414692 + 0.1112184370i\)

Euler product

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

−21.37816941214155871095672675016, −20.68777267313781384361979330199, −19.76033680280428346523347891416, −19.00919882825446699326889200703, −18.447209869006206674744110960368, −17.50636448399197412734524896897, −17.17993478235162094315311948564, −15.997936727994414557860126701160, −15.06761184659431458064794343175, −14.20319397117092750748564222672, −13.73231230788405701551737579079, −12.7353229391831979588370090156, −11.94529867073341269757384787244, −11.36213015003865587173152672071, −10.55302330165802215713653730177, −9.11784757732476169941618056181, −8.6055355914207777322132208689, −7.72772663237732441525837548578, −6.87920736583085787570551112506, −6.16374149270357520182516155621, −5.10247310585250079466958433572, −4.2706889393572705976970444677, −2.866831063978125196998524853615, −1.97700066736924765841193802516, −1.2223700552256733340641900298, 0.62524986528311629191561621388, 2.21402318800268668961887853204, 3.07509193281395887083103360931, 4.440769382203243926071700274949, 4.58406537526726868038998690668, 5.760106932583706573861850251041, 6.65982812343495181935477324516, 8.01840197362413139880303579256, 8.440850932089047126263541141602, 9.43386432722116368352685033983, 10.393563232694188263725388574303, 10.94385341490963460908802835511, 11.574772296276490243007961999285, 12.78480296122485998244809978037, 13.58851647556751684254598005065, 14.63176423012397436709833112784, 15.05810447098292894055090468300, 15.79546001041304238596081334540, 16.76042132931592053914578690495, 17.5020602527086729385090693790, 17.92528492273308422590303096886, 19.26020894256273416995539498036, 20.02670309754031591790787204124, 20.640655797420499560055940998969, 21.37085570993129255528111168154

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