Properties

Label 2-2200-1.1-c1-0-35
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.655·3-s + 0.415·7-s − 2.56·9-s + 11-s − 4·13-s + 6.51·17-s + 5.20·19-s − 0.272·21-s − 8.54·23-s + 3.65·27-s + 0.895·29-s − 6.73·31-s − 0.655·33-s − 8.96·37-s + 2.62·39-s + 10.0·41-s + 4.78·43-s − 5.61·47-s − 6.82·49-s − 4.27·51-s − 10.0·53-s − 3.41·57-s − 1.63·59-s + 7.10·61-s − 1.06·63-s − 10.6·67-s + 5.60·69-s + ⋯
L(s)  = 1  − 0.378·3-s + 0.157·7-s − 0.856·9-s + 0.301·11-s − 1.10·13-s + 1.58·17-s + 1.19·19-s − 0.0595·21-s − 1.78·23-s + 0.702·27-s + 0.166·29-s − 1.21·31-s − 0.114·33-s − 1.47·37-s + 0.420·39-s + 1.57·41-s + 0.730·43-s − 0.819·47-s − 0.975·49-s − 0.598·51-s − 1.37·53-s − 0.452·57-s − 0.212·59-s + 0.909·61-s − 0.134·63-s − 1.30·67-s + 0.674·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(17.5670\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 0.655T + 3T^{2} \)
7 \( 1 - 0.415T + 7T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 - 5.20T + 19T^{2} \)
23 \( 1 + 8.54T + 23T^{2} \)
29 \( 1 - 0.895T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 + 8.96T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + 1.63T + 59T^{2} \)
61 \( 1 - 7.10T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + 3.16T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 9.56T + 89T^{2} \)
97 \( 1 + 0.591T + 97T^{2} \)
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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

−8.646032211221730433941709440338, −7.74320881582948981197774628342, −7.31991338021486431889074864021, −6.07584260094966885907579237243, −5.57524344787703236403055972389, −4.81093360040561861007006999544, −3.64521479203511252444868787194, −2.80270801898650924544405580750, −1.52223992181878773680088038731, 0, 1.52223992181878773680088038731, 2.80270801898650924544405580750, 3.64521479203511252444868787194, 4.81093360040561861007006999544, 5.57524344787703236403055972389, 6.07584260094966885907579237243, 7.31991338021486431889074864021, 7.74320881582948981197774628342, 8.646032211221730433941709440338

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