Properties

Label 2-2200-1.1-c1-0-32
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 + 5.12·7-s + 3.56·9-s − 11-s − 3.12·13-s − 2·17-s − 4·19-s − 13.1·21-s − 6.56·23-s − 1.43·27-s + 3.12·29-s − 1.43·31-s + 2.56·33-s + 3.43·37-s + 8·39-s + 7.12·41-s − 1.12·43-s − 8·47-s + 19.2·49-s + 5.12·51-s + 4.24·53-s + 10.2·57-s − 12.8·59-s − 7.12·61-s + 18.2·63-s − 5.43·67-s + 16.8·69-s + ⋯
L(s)  = 1  − 1.47·3-s + 1.93·7-s + 1.18·9-s − 0.301·11-s − 0.866·13-s − 0.485·17-s − 0.917·19-s − 2.86·21-s − 1.36·23-s − 0.276·27-s + 0.579·29-s − 0.258·31-s + 0.445·33-s + 0.565·37-s + 1.28·39-s + 1.11·41-s − 0.171·43-s − 1.16·47-s + 2.74·49-s + 0.717·51-s + 0.583·53-s + 1.35·57-s − 1.66·59-s − 0.912·61-s + 2.29·63-s − 0.664·67-s + 2.02·69-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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 - 3.43T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 7.12T + 61T^{2} \)
67 \( 1 + 5.43T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 3.12T + 73T^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + 9.68T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.446064600014388309375931891414, −7.88784763572954475495488887979, −7.07837775051419882091553499224, −6.11840268150356585086802117767, −5.47666185808588562652194867158, −4.60767606098350367401776336190, −4.36757011990677979619389615173, −2.41191421436362608876075022357, −1.44965321046988511753828206052, 0, 1.44965321046988511753828206052, 2.41191421436362608876075022357, 4.36757011990677979619389615173, 4.60767606098350367401776336190, 5.47666185808588562652194867158, 6.11840268150356585086802117767, 7.07837775051419882091553499224, 7.88784763572954475495488887979, 8.446064600014388309375931891414

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