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Properties

Label 2-2200-440.109-c0-0-1
Degree $2$
Conductor $2200$
Sign $-0.447 - 0.894i$
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  + i·2-s − 4-s − i·8-s − 9-s + 11-s + i·13-s + 16-s − i·18-s + 19-s + i·22-s + i·23-s − 26-s + 29-s − 31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s − 9-s + 11-s + i·13-s + 16-s − i·18-s + 19-s + i·22-s + i·23-s − 26-s + 29-s − 31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9796492416\)
\(L(\frac12)\) \(\approx\) \(0.9796492416\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−9.375157805048728225412187538502, −8.685151635261986299317157540435, −7.909816433825972293531197398140, −7.10076469009060036209696039536, −6.38712039336697908439515981670, −5.70624140002490848807108576857, −4.85170682612017392845895388080, −3.94802294027652468214179499007, −3.05592497014781361296961078783, −1.36692376656180686332452212402, 0.77763693745920854202371468855, 2.17215931587145035144247821769, 3.15875523463681318037468542989, 3.79414088092177222857564009973, 4.98344026437409000481784922707, 5.57977441372606449807894571392, 6.56203591839882827670796680196, 7.67238946592889608100514947126, 8.591178154592207498301928381800, 8.903790632129862844993641713222

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