Properties

Label 2-2200-1.1-c1-0-43
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  + 1.56·3-s − 0.438·7-s − 0.561·9-s − 11-s − 7.12·13-s − 4.68·17-s + 5.56·19-s − 0.684·21-s + 7.12·23-s − 5.56·27-s + 4.43·29-s − 5.56·31-s − 1.56·33-s − 11.5·37-s − 11.1·39-s + 4.24·41-s − 5.12·43-s − 13.3·47-s − 6.80·49-s − 7.31·51-s + 2.68·53-s + 8.68·57-s − 7.12·59-s − 8.43·61-s + 0.246·63-s + 11.1·69-s − 8.68·71-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.165·7-s − 0.187·9-s − 0.301·11-s − 1.97·13-s − 1.13·17-s + 1.27·19-s − 0.149·21-s + 1.48·23-s − 1.07·27-s + 0.824·29-s − 0.998·31-s − 0.271·33-s − 1.90·37-s − 1.78·39-s + 0.663·41-s − 0.781·43-s − 1.95·47-s − 0.972·49-s − 1.02·51-s + 0.368·53-s + 1.15·57-s − 0.927·59-s − 1.08·61-s + 0.0310·63-s + 1.33·69-s − 1.03·71-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 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
13 \( 1 + 7.12T + 13T^{2} \)
17 \( 1 + 4.68T + 17T^{2} \)
19 \( 1 - 5.56T + 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 - 2.68T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 8.43T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 8.68T + 71T^{2} \)
73 \( 1 - 7.12T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 2.68T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.810396538863175355179999383484, −7.84392484829148481377827725106, −7.28109951824093198299008720832, −6.53999919205739018185548424832, −5.16380139545867065441953244621, −4.84019275934143573144119954468, −3.37702356766540203005667795978, −2.83740340563170843539856462838, −1.88546454629694874648105451316, 0, 1.88546454629694874648105451316, 2.83740340563170843539856462838, 3.37702356766540203005667795978, 4.84019275934143573144119954468, 5.16380139545867065441953244621, 6.53999919205739018185548424832, 7.28109951824093198299008720832, 7.84392484829148481377827725106, 8.810396538863175355179999383484

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