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Properties

Label 2-2200-440.339-c0-0-0
Degree $2$
Conductor $2200$
Sign $-0.487 - 0.873i$
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  + (−0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 + 0.587i)4-s + (0.190 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.809 + 0.587i)11-s − 0.618i·12-s + (0.309 + 0.951i)16-s + (−1.53 + 0.5i)17-s + (−0.363 + 0.5i)18-s + (−1.30 + 0.951i)19-s + (0.951 − 0.309i)22-s + (−0.190 + 0.587i)24-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 + 0.587i)4-s + (0.190 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.809 + 0.587i)11-s − 0.618i·12-s + (0.309 + 0.951i)16-s + (−1.53 + 0.5i)17-s + (−0.363 + 0.5i)18-s + (−1.30 + 0.951i)19-s + (0.951 − 0.309i)22-s + (−0.190 + 0.587i)24-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1049233629\)
\(L(\frac12)\) \(\approx\) \(0.1049233629\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.542338919129069060334444182985, −8.643219651981551172743331314919, −8.097119214554545959728556701912, −7.18955071334825375282724319560, −6.55547059240761230859432132376, −5.96127975563459250963328567510, −4.57505951076480934928119117675, −3.64918584868202388019641800559, −2.37469161025213237139361053451, −1.60319146695806167527888023000, 0.095898501228882808080165206418, 1.99883720124931114756854486186, 2.81465784407975645527680044365, 4.38662802393577423252375762693, 5.05836847743204466521360902089, 5.95852711117514214995997992905, 6.74455379809759850603467203065, 7.50409371353508635602436388759, 8.368882857686720038460080490847, 8.922397767328054608188057937346

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