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Properties

Label 1-1100-1100.359-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.992 + 0.118i$
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  + 3-s + (0.809 − 0.587i)7-s + 9-s + 13-s + (−0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + 27-s + (−0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.309 − 0.951i)37-s + 39-s + (0.809 − 0.587i)41-s − 43-s + ⋯
L(s)  = 1  + 3-s + (0.809 − 0.587i)7-s + 9-s + 13-s + (−0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + 27-s + (−0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.309 − 0.951i)37-s + 39-s + (0.809 − 0.587i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.992 + 0.118i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.992 + 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.598698665 + 0.1547557679i\)
\(L(\frac12)\) \(\approx\) \(2.598698665 + 0.1547557679i\)
\(L(1)\) \(\approx\) \(1.699090020 + 0.02065491986i\)
\(L(1)\) \(\approx\) \(1.699090020 + 0.02065491986i\)

Euler product

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

−21.23641702688559203270590777036, −20.55263155896770357697172807805, −20.068572357342768655939998670350, −18.96730795097313364155657786849, −18.40512584247124180423402507665, −17.78471455056613545783796243568, −16.64379890994186192859843997560, −15.596584213673112846527920420910, −15.23295157443121375446612783398, −14.37855709526552296657589587509, −13.50403003434011167553810714373, −13.06674414203119929545625335042, −11.80004319439477158292812617379, −11.18293077836712728378581879585, −10.16903652921174311718114967111, −9.070191751114856121191504701656, −8.711908043906675916144929820630, −7.843817417800386696464814178566, −6.97528721147657787989525855116, −5.9549372654138218410354177774, −4.770082169887554635860779426968, −4.11932526933003561476438708971, −2.86906003264315744452266037287, −2.24366533976841653494467174625, −1.11966318381965513819481567904, 1.3469159838607421726407001057, 1.89145278608546136484857578158, 3.33579190653606179584381753127, 3.881678193165509335321654168143, 4.854835723750988408091303340395, 5.982307720328150148895490866646, 7.16037898954734077332279752430, 7.72852255417619478816655444054, 8.67352815684550939621323037958, 9.154354724113141054850155345716, 10.47676698859064615001769916764, 10.825142050600787976383383638544, 12.02423175326211718979209994223, 12.99263521490339607293157080979, 13.71623916694670754696031702883, 14.29124422972006029370278291115, 15.08707661438556376680697534828, 15.84505098052039321547326014780, 16.68144283654053120720701526217, 17.77859948520739678062300364186, 18.27606876613493009668085030225, 19.24618615019767507355888804226, 19.98682679288677870346589681911, 20.59314782796455602622744663770, 21.25492189002283489547949906742

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