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Properties

Label 2-2200-440.283-c0-0-7
Degree $2$
Conductor $2200$
Sign $0.191 + 0.981i$
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 + (0.903 − 1.77i)3-s + (0.951 + 0.309i)4-s + (1.16 − 1.60i)6-s + (0.891 + 0.453i)8-s + (−1.73 − 2.39i)9-s + (−0.978 − 0.207i)11-s + (1.40 − 1.40i)12-s + (0.809 + 0.587i)16-s + (0.209 + 1.32i)17-s + (−1.34 − 2.63i)18-s + (0.128 + 0.395i)19-s + (−0.933 − 0.358i)22-s + (1.60 − 1.16i)24-s + (−3.84 + 0.608i)27-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.903 − 1.77i)3-s + (0.951 + 0.309i)4-s + (1.16 − 1.60i)6-s + (0.891 + 0.453i)8-s + (−1.73 − 2.39i)9-s + (−0.978 − 0.207i)11-s + (1.40 − 1.40i)12-s + (0.809 + 0.587i)16-s + (0.209 + 1.32i)17-s + (−1.34 − 2.63i)18-s + (0.128 + 0.395i)19-s + (−0.933 − 0.358i)22-s + (1.60 − 1.16i)24-s + (−3.84 + 0.608i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.191 + 0.981i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (2043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.191 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.732778843\)
\(L(\frac12)\) \(\approx\) \(2.732778843\)
\(L(1)\) 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 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (-0.903 + 1.77i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.209 - 1.32i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.128 - 0.395i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.773 + 0.251i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.05 + 1.05i)T - iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.829 - 1.62i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.80 - 0.285i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 - 0.209iT - T^{2} \)
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

−8.500006334289555271965727904198, −8.119811123034102454059627268544, −7.50636428653522944913676183141, −6.71808129542712886952608416905, −6.06914014434056068630023790438, −5.37442817419625372563684281631, −3.91340373267169750946979215889, −3.11659893065665098248978976840, −2.31807156426025532501395239714, −1.44113022485279082979228309266, 2.33071995671389383658735684191, 2.91342778770304329960583486175, 3.69927491331631794719893479114, 4.63628609873363049461962185330, 5.06663155242659950745758683989, 5.78613392676489534077406995757, 7.20218952809645276095315286198, 7.87654890766350057236121710405, 8.797368473105708938666096283523, 9.637409521295186822248295354906

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