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Properties

Label 2-2200-440.123-c0-0-2
Degree $2$
Conductor $2200$
Sign $0.537 - 0.842i$
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  + (−0.891 + 0.453i)2-s + (1.87 + 0.297i)3-s + (0.587 − 0.809i)4-s + (−1.80 + 0.587i)6-s + (−0.156 + 0.987i)8-s + (2.48 + 0.809i)9-s + (−0.809 + 0.587i)11-s + (1.34 − 1.34i)12-s + (−0.309 − 0.951i)16-s + (−0.734 + 1.44i)17-s + (−2.58 + 0.409i)18-s + (0.951 − 0.690i)19-s + (0.453 − 0.891i)22-s + (−0.587 + 1.80i)24-s + (2.74 + 1.39i)27-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (1.87 + 0.297i)3-s + (0.587 − 0.809i)4-s + (−1.80 + 0.587i)6-s + (−0.156 + 0.987i)8-s + (2.48 + 0.809i)9-s + (−0.809 + 0.587i)11-s + (1.34 − 1.34i)12-s + (−0.309 − 0.951i)16-s + (−0.734 + 1.44i)17-s + (−2.58 + 0.409i)18-s + (0.951 − 0.690i)19-s + (0.453 − 0.891i)22-s + (−0.587 + 1.80i)24-s + (2.74 + 1.39i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.537 - 0.842i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.537 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.491169851\)
\(L(\frac12)\) \(\approx\) \(1.491169851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 \)
11 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-1.87 - 0.297i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.734 - 1.44i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.610 + 0.0966i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.550 - 0.280i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 + 0.618iT - T^{2} \)
97 \( 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2} \)
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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

−9.163030958513764403452064010707, −8.581336065770596016292594336090, −8.029545263639532835998919313994, −7.33345460492962293455929044780, −6.72593695183336426851181294176, −5.38780588883572389101123933359, −4.46075608750883144904974753912, −3.39130098993564182192355869856, −2.40554666552185192798824642789, −1.71767791877256923567708553645, 1.25344506599127248424003175641, 2.40905573807303166803956225299, 2.97746802790216481413042156641, 3.68941191585827435569773861744, 4.87564311090182260319525677081, 6.44138188582422118649240446344, 7.28860017740730530481345551378, 7.81890431731123266485795942811, 8.386097190511226544366429122602, 9.083974067696864073374751482680

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