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Properties

Label 1-1100-1100.411-r1-0-0
Degree $1$
Conductor $1100$
Sign $-0.163 - 0.986i$
Analytic cond. $118.211$
Root an. cond. $118.211$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(118.211\)
Root analytic conductor: \(118.211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (1:\ ),\ -0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9748949518 - 1.150326804i\)
\(L(\frac12)\) \(\approx\) \(0.9748949518 - 1.150326804i\)
\(L(1)\) \(\approx\) \(0.9040960802 - 0.3143116327i\)
\(L(1)\) \(\approx\) \(0.9040960802 - 0.3143116327i\)

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

−21.42072551182583372599870086693, −20.782733281686159172553489670174, −20.12272252287581281331596789839, −19.108629738373276544175085268844, −18.16311610193382996341605534279, −17.479020185662708787749322064202, −16.79508870996346172478329042129, −15.92034119545035101264402876247, −15.13949914968025523003080699117, −14.64092459425432050373967805033, −13.81178734079645212082082670220, −12.46598032040437972931508442760, −11.911659863149820367576059910343, −11.09256000067353989325420221195, −10.259989170099678739648597444401, −9.60779778332407246461366200351, −8.56882937039656482594371510015, −8.01436200988890771481380944677, −6.68544362905390263929497033717, −5.78023153874437464715886468447, −4.83934066613124618026810511238, −4.478671096909131281475146927359, −3.06194916156467018957216487081, −2.34359511232046793401741392504, −0.77429867923495907827264970602, 0.42186868494035968538088852238, 1.59642962096108542195274632594, 2.18075413708947901624027913343, 3.60077980357850340094787149897, 4.668829674089997105396135688978, 5.47598039359000593493535026297, 6.56819628761933336085940526144, 7.16000220148889119219543024928, 8.09815376566073258681071210888, 8.633800119721879538793866617849, 10.04801907806212257388565326970, 10.7402169619387198871188948621, 11.66857466718587818149812885375, 12.23887179405536651789214605584, 13.11274207213798358801472926582, 14.00035914107765640171035952694, 14.46376299310290830597063242954, 15.47987511663274064033164163513, 16.69846913935619669038606475002, 17.26348351590734477422551325693, 17.72366067473693820757439601995, 18.70893306477671670109593441726, 19.55026925745934534283665190684, 19.852961414462783015478711648982, 21.27136090407364144946577688647

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