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Properties

Label 2-2200-440.179-c0-0-0
Degree $2$
Conductor $2200$
Sign $0.855 - 0.518i$
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 + (−0.198 + 0.0646i)3-s + (−0.309 + 0.951i)4-s + (0.169 + 0.122i)6-s + (0.951 − 0.309i)8-s + (−0.773 + 0.562i)9-s + (−0.978 − 0.207i)11-s − 0.209i·12-s + (−0.809 − 0.587i)16-s + (0.786 − 1.08i)17-s + (0.909 + 0.295i)18-s + (0.604 + 1.86i)19-s + (0.406 + 0.913i)22-s + (−0.169 + 0.122i)24-s + (0.240 − 0.330i)27-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.198 + 0.0646i)3-s + (−0.309 + 0.951i)4-s + (0.169 + 0.122i)6-s + (0.951 − 0.309i)8-s + (−0.773 + 0.562i)9-s + (−0.978 − 0.207i)11-s − 0.209i·12-s + (−0.809 − 0.587i)16-s + (0.786 − 1.08i)17-s + (0.909 + 0.295i)18-s + (0.604 + 1.86i)19-s + (0.406 + 0.913i)22-s + (−0.169 + 0.122i)24-s + (0.240 − 0.330i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.855 - 0.518i$
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: \((2,\ 2200,\ (\ :0),\ 0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5965834616\)
\(L(\frac12)\) \(\approx\) \(0.5965834616\)
\(L(1)\) 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 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (0.198 - 0.0646i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.786 + 1.08i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.604 - 1.86i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 - 1.33iT - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.73 + 0.564i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.07 - 1.47i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.209T + T^{2} \)
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

−9.532348554622145947474867621166, −8.497027448049896774819561973795, −7.86777061570951827969642300278, −7.45406579103164509002849188109, −6.01475857506019992135662902728, −5.30631124477720186083109368894, −4.39094009789646853118657296097, −3.16277340597896095721581319991, −2.64736107922686510646983411470, −1.25942125968854472965533328277, 0.56938924901946967349120053388, 2.16668286960310251652823614702, 3.38180290810861277527134250458, 4.65930525024339810500309384626, 5.50559200536896774280629064692, 5.96304849064953291943207751761, 7.06135079687165030430293838754, 7.47406224782712388923281811486, 8.564973676107898444642959986366, 8.883260626027405415325989531448

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