Properties

Label 2-2200-1.1-c1-0-40
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 − 3.56·7-s − 0.561·9-s + 11-s + 3.12·13-s − 5.56·17-s + 2.43·19-s − 5.56·21-s − 7.12·23-s − 5.56·27-s − 0.438·29-s + 8.68·31-s + 1.56·33-s − 9.80·37-s + 4.87·39-s − 10·41-s − 5.12·43-s + 7.12·47-s + 5.68·49-s − 8.68·51-s + 4.43·53-s + 3.80·57-s − 13.3·59-s − 3.56·61-s + 2·63-s − 11.1·69-s − 2.43·71-s + ⋯
L(s)  = 1  + 0.901·3-s − 1.34·7-s − 0.187·9-s + 0.301·11-s + 0.866·13-s − 1.34·17-s + 0.559·19-s − 1.21·21-s − 1.48·23-s − 1.07·27-s − 0.0814·29-s + 1.55·31-s + 0.271·33-s − 1.61·37-s + 0.780·39-s − 1.56·41-s − 0.781·43-s + 1.03·47-s + 0.812·49-s − 1.21·51-s + 0.609·53-s + 0.504·57-s − 1.74·59-s − 0.456·61-s + 0.251·63-s − 1.33·69-s − 0.289·71-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 \( 1 + 3.56T + 7T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
23 \( 1 + 7.12T + 23T^{2} \)
29 \( 1 + 0.438T + 29T^{2} \)
31 \( 1 - 8.68T + 31T^{2} \)
37 \( 1 + 9.80T + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 3.56T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 2.43T + 71T^{2} \)
73 \( 1 + 4.87T + 73T^{2} \)
79 \( 1 - 0.876T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.664844501141892481235139856392, −8.168690694091231830047867080055, −7.02098137938668714009051259072, −6.41118113912557275957496763110, −5.69398656639830046496126811810, −4.34960802384987719795808586581, −3.52029124049328955310852894781, −2.90828889456437385287581801841, −1.81062257481611373367775304370, 0, 1.81062257481611373367775304370, 2.90828889456437385287581801841, 3.52029124049328955310852894781, 4.34960802384987719795808586581, 5.69398656639830046496126811810, 6.41118113912557275957496763110, 7.02098137938668714009051259072, 8.168690694091231830047867080055, 8.664844501141892481235139856392

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