Properties

Label 2-2200-440.227-c0-0-3
Degree $2$
Conductor $2200$
Sign $0.888 - 0.458i$
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  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (−0.280 + 0.550i)7-s + (−0.453 − 0.891i)8-s + (0.587 − 0.809i)9-s + (−0.309 − 0.951i)11-s + (1.59 − 0.253i)13-s + (−0.587 − 0.190i)14-s + (0.809 − 0.587i)16-s + (0.891 + 0.453i)18-s + (0.363 − 1.11i)19-s + (0.891 − 0.453i)22-s + (−1.34 − 1.34i)23-s + (0.5 + 1.53i)26-s + (0.0966 − 0.610i)28-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (−0.280 + 0.550i)7-s + (−0.453 − 0.891i)8-s + (0.587 − 0.809i)9-s + (−0.309 − 0.951i)11-s + (1.59 − 0.253i)13-s + (−0.587 − 0.190i)14-s + (0.809 − 0.587i)16-s + (0.891 + 0.453i)18-s + (0.363 − 1.11i)19-s + (0.891 − 0.453i)22-s + (−1.34 − 1.34i)23-s + (0.5 + 1.53i)26-s + (0.0966 − 0.610i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152115491\)
\(L(\frac12)\) \(\approx\) \(1.152115491\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
5 \( 1 \)
11 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.280 - 0.550i)T + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.04 - 0.533i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.533 - 1.04i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.297 - 1.87i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + 0.618iT - T^{2} \)
97 \( 1 + (-0.951 + 0.309i)T^{2} \)
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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

−8.907525370933474339120638034357, −8.623474679019507690900116536027, −7.78958051807034868506419766562, −6.75001025861789888413187346721, −6.16699536473097561399273445437, −5.68085832418690579706790838456, −4.50787345578506883387474044352, −3.71891079656527404163483033365, −2.84909919612799830037712047324, −0.879126985253945424015587541845, 1.42836479715285732557789488873, 2.13900383238298307393073853555, 3.73001320708557504630139335024, 3.88850847905997509708192749129, 5.07270162999177673574075033897, 5.80475511797025434205700237769, 6.89090006376065853032648259376, 7.86160409814727473599406707893, 8.395114099612429640959324233840, 9.518077649132806721064442561108

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