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Properties

Label 2-2200-440.179-c0-0-4
Degree $2$
Conductor $2200$
Sign $0.481 - 0.876i$
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  + (0.587 + 0.809i)2-s + (1.53 − 0.5i)3-s + (−0.309 + 0.951i)4-s + (1.30 + 0.951i)6-s + (−0.951 + 0.309i)8-s + (1.30 − 0.951i)9-s + (0.309 + 0.951i)11-s + 1.61i·12-s + (−0.809 − 0.587i)16-s + (−0.363 + 0.5i)17-s + (1.53 + 0.5i)18-s + (−0.190 − 0.587i)19-s + (−0.587 + 0.809i)22-s + (−1.30 + 0.951i)24-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (1.53 − 0.5i)3-s + (−0.309 + 0.951i)4-s + (1.30 + 0.951i)6-s + (−0.951 + 0.309i)8-s + (1.30 − 0.951i)9-s + (0.309 + 0.951i)11-s + 1.61i·12-s + (−0.809 − 0.587i)16-s + (−0.363 + 0.5i)17-s + (1.53 + 0.5i)18-s + (−0.190 − 0.587i)19-s + (−0.587 + 0.809i)22-s + (−1.30 + 0.951i)24-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.486814633\)
\(L(\frac12)\) \(\approx\) \(2.486814633\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
11 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 0.618iT - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
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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

−8.882139499286473315635951262240, −8.662577128067470986923895180670, −7.64507934839079107698513442912, −7.18757678258621547580098510986, −6.54891325395236333381930146083, −5.43168918403594551742370477966, −4.34190040218911784525371077216, −3.74620694172068306280437833649, −2.72021820666809151789294472924, −1.91133873041091227271823132705, 1.47620738557601223109798294747, 2.63018768543741175759583711840, 3.20924825979449144550501972509, 4.00127692646093958491185285143, 4.71269353877652437325620627949, 5.79452838723291860874337132842, 6.69784415887523418697987711735, 7.900684660933683309390543622326, 8.559302469865155755567275560608, 9.206178233503060604798518996270

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