Properties

Label 2-2200-1.1-c1-0-25
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  − 2.56·3-s + 0.561·7-s + 3.56·9-s + 11-s − 5.12·13-s − 1.43·17-s + 6.56·19-s − 1.43·21-s + 1.12·23-s − 1.43·27-s − 4.56·29-s − 3.68·31-s − 2.56·33-s + 10.8·37-s + 13.1·39-s − 10·41-s + 3.12·43-s − 1.12·47-s − 6.68·49-s + 3.68·51-s + 8.56·53-s − 16.8·57-s + 11.3·59-s + 0.561·61-s + 2.00·63-s − 2.87·69-s − 6.56·71-s + ⋯
L(s)  = 1  − 1.47·3-s + 0.212·7-s + 1.18·9-s + 0.301·11-s − 1.42·13-s − 0.348·17-s + 1.50·19-s − 0.313·21-s + 0.234·23-s − 0.276·27-s − 0.847·29-s − 0.661·31-s − 0.445·33-s + 1.77·37-s + 2.10·39-s − 1.56·41-s + 0.476·43-s − 0.163·47-s − 0.954·49-s + 0.515·51-s + 1.17·53-s − 2.22·57-s + 1.48·59-s + 0.0718·61-s + 0.251·63-s − 0.346·69-s − 0.778·71-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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 - 0.561T + 7T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + 1.43T + 17T^{2} \)
19 \( 1 - 6.56T + 19T^{2} \)
23 \( 1 - 1.12T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + 3.68T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 3.12T + 43T^{2} \)
47 \( 1 + 1.12T + 47T^{2} \)
53 \( 1 - 8.56T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 0.561T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 6.56T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 8.87T + 97T^{2} \)
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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

−8.754326514318178105048230077046, −7.53835886051522739675359849153, −7.13212239517330728218967992933, −6.22068970411549740739863278979, −5.36260281926074019882784725267, −4.96670183560935242326581246944, −3.96971013751382362428356247847, −2.65312902819391403683713662019, −1.28905278132485924618995987126, 0, 1.28905278132485924618995987126, 2.65312902819391403683713662019, 3.96971013751382362428356247847, 4.96670183560935242326581246944, 5.36260281926074019882784725267, 6.22068970411549740739863278979, 7.13212239517330728218967992933, 7.53835886051522739675359849153, 8.754326514318178105048230077046

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