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Properties

Label 2-2200-440.227-c0-0-1
Degree $2$
Conductor $2200$
Sign $0.151 - 0.988i$
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  + (−0.987 + 0.156i)2-s + (0.533 + 1.04i)3-s + (0.951 − 0.309i)4-s + (−0.690 − 0.951i)6-s + (−0.891 + 0.453i)8-s + (−0.224 + 0.309i)9-s + (0.309 − 0.951i)11-s + (0.831 + 0.831i)12-s + (0.809 − 0.587i)16-s + (−0.0966 + 0.610i)17-s + (0.173 − 0.340i)18-s + (−0.587 + 1.80i)19-s + (−0.156 + 0.987i)22-s + (−0.951 − 0.690i)24-s + (0.717 + 0.113i)27-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (0.533 + 1.04i)3-s + (0.951 − 0.309i)4-s + (−0.690 − 0.951i)6-s + (−0.891 + 0.453i)8-s + (−0.224 + 0.309i)9-s + (0.309 − 0.951i)11-s + (0.831 + 0.831i)12-s + (0.809 − 0.587i)16-s + (−0.0966 + 0.610i)17-s + (0.173 − 0.340i)18-s + (−0.587 + 1.80i)19-s + (−0.156 + 0.987i)22-s + (−0.951 − 0.690i)24-s + (0.717 + 0.113i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9324097395\)
\(L(\frac12)\) \(\approx\) \(0.9324097395\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 \)
11 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.533 - 1.04i)T + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.0966 - 0.610i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.734 + 1.44i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.59 + 0.253i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + 1.61iT - T^{2} \)
97 \( 1 + (0.183 + 1.16i)T + (-0.951 + 0.309i)T^{2} \)
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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

−9.466303886642269004168879629250, −8.605790691046365498496356069537, −8.255775131374505162290662362077, −7.34989391336810980447003651997, −6.16112458116312509273082760866, −5.81714017475122616378669067429, −4.38078050133908711254529784787, −3.61553367513189771881010171137, −2.71464484686047600355835233130, −1.37329979031761905396068689868, 0.932868461372517207951217022306, 2.22276026339679738178763039515, 2.57034191916691834435799917072, 3.99434410529297291069974504436, 5.16814889954468816435696861621, 6.51182778772281583646669930078, 6.98264559987894376649240748460, 7.50369813462896910101057772219, 8.288692983944717929740831327585, 9.081063832071268351908314930892

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