Properties

Label 2-2200-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.841·3-s + 3.29·7-s − 2.29·9-s + 11-s − 13-s + 1.15·17-s + 1.31·19-s − 2.76·21-s − 3.84·23-s + 4.45·27-s + 6.61·29-s + 7.58·31-s − 0.841·33-s − 2.13·37-s + 0.841·39-s + 6.13·41-s − 8.81·43-s − 2.76·47-s + 3.84·49-s − 0.974·51-s + 10.1·53-s − 1.10·57-s + 2.76·59-s − 3.61·61-s − 7.54·63-s − 2.90·67-s + 3.23·69-s + ⋯
L(s)  = 1  − 0.485·3-s + 1.24·7-s − 0.764·9-s + 0.301·11-s − 0.277·13-s + 0.281·17-s + 0.302·19-s − 0.604·21-s − 0.800·23-s + 0.856·27-s + 1.22·29-s + 1.36·31-s − 0.146·33-s − 0.350·37-s + 0.134·39-s + 0.957·41-s − 1.34·43-s − 0.403·47-s + 0.548·49-s − 0.136·51-s + 1.40·53-s − 0.146·57-s + 0.360·59-s − 0.462·61-s − 0.951·63-s − 0.354·67-s + 0.388·69-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: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.623107155\)
\(L(\frac12)\) \(\approx\) \(1.623107155\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.841T + 3T^{2} \)
7 \( 1 - 3.29T + 7T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 1.15T + 17T^{2} \)
19 \( 1 - 1.31T + 19T^{2} \)
23 \( 1 + 3.84T + 23T^{2} \)
29 \( 1 - 6.61T + 29T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + 2.13T + 37T^{2} \)
41 \( 1 - 6.13T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 + 2.76T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 2.76T + 59T^{2} \)
61 \( 1 + 3.61T + 61T^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 - 0.866T + 71T^{2} \)
73 \( 1 + 5.87T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 8.92T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.855407846754589251952771130249, −8.289977542696076268280963756645, −7.63984965333184882029925758030, −6.60468059558720731752068743228, −5.87041131645338975532760063743, −5.02429193496024749111496118442, −4.45372340377730391920079787972, −3.20000392612341435952889466879, −2.10938615306349937776964465552, −0.874525955513996168433282289909, 0.874525955513996168433282289909, 2.10938615306349937776964465552, 3.20000392612341435952889466879, 4.45372340377730391920079787972, 5.02429193496024749111496118442, 5.87041131645338975532760063743, 6.60468059558720731752068743228, 7.63984965333184882029925758030, 8.289977542696076268280963756645, 8.855407846754589251952771130249

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