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Properties

Label 2-2200-88.27-c0-0-5
Degree $2$
Conductor $2200$
Sign $-0.0457 + 0.998i$
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  + (−0.309 − 0.951i)2-s + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.309 − 0.951i)16-s + (0.5 − 1.53i)17-s + (−0.5 + 0.363i)18-s + (1.30 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.190 + 0.587i)24-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.309 − 0.951i)16-s + (0.5 − 1.53i)17-s + (−0.5 + 0.363i)18-s + (1.30 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.190 + 0.587i)24-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.0457 + 0.998i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (2051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021246536\)
\(L(\frac12)\) \(\approx\) \(1.021246536\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + 0.618T + T^{2} \)
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 \( 1 - 1.61T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.247650372337260065120472888611, −8.399367784788583336861022788743, −7.85658230835371947491012282230, −6.93600964844870759520643863493, −5.54759667591958564634008288264, −4.99173590485576578852466093235, −3.68030398588449833708107847547, −3.25244401850014880466607744765, −2.35828107882529096467711899096, −0.822531400857455061362497654972, 1.42380497132462276101063569383, 2.62145598071620995688868743109, 3.82510869626105543069558411739, 4.98593204750475581071293051266, 5.45094109738482224200887376974, 6.50144902097619749598806099721, 7.33019955386800141383572182915, 7.85782329058285093430648930201, 8.428896677191885020698216158774, 9.217282875419131376557873854181

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