Properties

Label 2-2200-1.1-c1-0-21
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  − 3.36·3-s − 1.08·7-s + 8.29·9-s + 11-s − 4·13-s + 0.107·17-s − 6.61·19-s + 3.63·21-s + 5.97·23-s − 17.7·27-s + 7.80·29-s + 1.12·31-s − 3.36·33-s + 7.05·37-s + 13.4·39-s + 5.19·41-s − 5.52·43-s + 7.69·47-s − 5.82·49-s − 0.362·51-s + 4.77·53-s + 22.2·57-s − 0.677·59-s + 0.197·61-s − 8.97·63-s + 1.41·67-s − 20.0·69-s + ⋯
L(s)  = 1  − 1.93·3-s − 0.409·7-s + 2.76·9-s + 0.301·11-s − 1.10·13-s + 0.0261·17-s − 1.51·19-s + 0.793·21-s + 1.24·23-s − 3.42·27-s + 1.44·29-s + 0.202·31-s − 0.584·33-s + 1.15·37-s + 2.15·39-s + 0.810·41-s − 0.843·43-s + 1.12·47-s − 0.832·49-s − 0.0507·51-s + 0.656·53-s + 2.94·57-s − 0.0882·59-s + 0.0252·61-s − 1.13·63-s + 0.173·67-s − 2.41·69-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 \( 1 + 3.36T + 3T^{2} \)
7 \( 1 + 1.08T + 7T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 0.107T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 5.52T + 43T^{2} \)
47 \( 1 - 7.69T + 47T^{2} \)
53 \( 1 - 4.77T + 53T^{2} \)
59 \( 1 + 0.677T + 59T^{2} \)
61 \( 1 - 0.197T + 61T^{2} \)
67 \( 1 - 1.41T + 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 + 6.16T + 73T^{2} \)
79 \( 1 + 9.35T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 - 6.60T + 97T^{2} \)
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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

−8.781497194541708259235397294642, −7.57119704783263977508121221574, −6.80630808319915914153503474575, −6.36697357534299642309768864298, −5.54600712040268113191777841465, −4.69351417183543095157122392324, −4.20518472634716369482539319664, −2.62250040110565721775495980004, −1.16733172319793113448250980886, 0, 1.16733172319793113448250980886, 2.62250040110565721775495980004, 4.20518472634716369482539319664, 4.69351417183543095157122392324, 5.54600712040268113191777841465, 6.36697357534299642309768864298, 6.80630808319915914153503474575, 7.57119704783263977508121221574, 8.781497194541708259235397294642

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