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Properties

Label 1-1100-1100.611-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.895 + 0.445i$
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 + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s − 17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.309 + 0.951i)23-s + (0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + 37-s + (−0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)13-s − 17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.309 + 0.951i)23-s + (0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + 37-s + (−0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.895 + 0.445i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255503185 + 0.2951656337i\)
\(L(\frac12)\) \(\approx\) \(1.255503185 + 0.2951656337i\)
\(L(1)\) \(\approx\) \(0.9701107481 + 0.1835568927i\)
\(L(1)\) \(\approx\) \(0.9701107481 + 0.1835568927i\)

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 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + T \)
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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

−21.42411746324643012344550616832, −20.47890113082290814238178774292, −19.725073832405683983760642055128, −18.84099981116413899241605316715, −18.196802931610953996468092095973, −17.795303703002614153849500560831, −16.76031371469785568683717651497, −15.965967373913047426371275770244, −15.00665849055532231558320061104, −14.32462046958100341801408474510, −13.2404849595271872036978549361, −12.75230903737342443435813517425, −11.94499859857924312282167722238, −11.15425301891598709019628452515, −10.476366359089996723457963931259, −9.07535914826136594987033748407, −8.40006787157452478871591489443, −7.78482773369455753580967878819, −6.509635171605405879531905517279, −6.08150241641248476207286040402, −5.17142784537668155494071452473, −4.06939366645460705346963916480, −2.64260937497953837669989905954, −2.079346675932842417630991398526, −0.83735807872905514201552521742, 0.79506203606226513720467850486, 2.18489449095211303352410243774, 3.49305777507431827559029127607, 4.24675758117435824997820496571, 4.80620517162647954333016793076, 6.04068726210546726636487582701, 6.71404572012136742915703812251, 7.85050264535107133205978758741, 8.83741741145590534933991310650, 9.47453431000086772843997276153, 10.55984166129647711950339300640, 10.99704849893910079945674614243, 11.67594445111780047662743361365, 12.88354904203563504686222731051, 13.8009203508822795394838745896, 14.3700812327821163927716463577, 15.46513311952468239416387912716, 15.93063411759987190263419277850, 16.81332988271617895317240544300, 17.50948521962702636773913017770, 18.05891190398067405544612066722, 19.46870393805852193241100805145, 19.91483018418685781136839061842, 20.90853473990171349254882765503, 21.3494750077474274231749414997

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