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Properties

Label 2-2200-440.283-c0-0-4
Degree $2$
Conductor $2200$
Sign $-0.151 - 0.988i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
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  + (0.987 + 0.156i)2-s + (−0.533 + 1.04i)3-s + (0.951 + 0.309i)4-s + (−0.690 + 0.951i)6-s + (0.891 + 0.453i)8-s + (−0.224 − 0.309i)9-s + (0.309 + 0.951i)11-s + (−0.831 + 0.831i)12-s + (0.809 + 0.587i)16-s + (0.0966 + 0.610i)17-s + (−0.173 − 0.340i)18-s + (−0.587 − 1.80i)19-s + (0.156 + 0.987i)22-s + (−0.951 + 0.690i)24-s + (−0.717 + 0.113i)27-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (−0.533 + 1.04i)3-s + (0.951 + 0.309i)4-s + (−0.690 + 0.951i)6-s + (0.891 + 0.453i)8-s + (−0.224 − 0.309i)9-s + (0.309 + 0.951i)11-s + (−0.831 + 0.831i)12-s + (0.809 + 0.587i)16-s + (0.0966 + 0.610i)17-s + (−0.173 − 0.340i)18-s + (−0.587 − 1.80i)19-s + (0.156 + 0.987i)22-s + (−0.951 + 0.690i)24-s + (−0.717 + 0.113i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.151 - 0.988i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (2043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.902660207\)
\(L(\frac12)\) \(\approx\) \(1.902660207\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
5 \( 1 \)
11 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.533 - 1.04i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.0966 - 0.610i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.14 - 1.14i)T - iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.734 + 1.44i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 - 1.61iT - T^{2} \)
97 \( 1 + (-0.183 + 1.16i)T + (-0.951 - 0.309i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.609976328895164712305070313856, −8.772733635640874224611383963093, −7.66869356269300838361313015174, −6.91632536062406395576451719036, −6.16064876929730431942512171710, −5.26688613171763077597186973652, −4.55076886512771157201431613099, −4.16434728069568127477905365182, −3.02474338716133891708558498882, −1.89986989886309550856425530839, 1.10823610230029181300690560358, 2.08621020857600811656608184924, 3.29465083595629117333908357502, 4.09033106382829836777728888845, 5.22990038204770958009810510797, 6.10262798051114302566360008773, 6.29765798031946913485116626451, 7.36245849489391571171170578821, 7.87453569119421969631996959852, 8.950644838522866008594393142555

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