Properties

Label 2-2200-1.1-c3-0-106
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $129.804$
Root an. cond. $11.3931$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 8·7-s − 23·9-s − 11·11-s − 25·13-s + 52·17-s + 19·19-s − 16·21-s + 75·23-s − 100·27-s + 241·29-s + 189·31-s − 22·33-s − 68·37-s − 50·39-s + 240·41-s + 183·43-s + 56·47-s − 279·49-s + 104·51-s + 142·53-s + 38·57-s − 584·59-s − 622·61-s + 184·63-s + 142·67-s + 150·69-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.431·7-s − 0.851·9-s − 0.301·11-s − 0.533·13-s + 0.741·17-s + 0.229·19-s − 0.166·21-s + 0.679·23-s − 0.712·27-s + 1.54·29-s + 1.09·31-s − 0.116·33-s − 0.302·37-s − 0.205·39-s + 0.914·41-s + 0.649·43-s + 0.173·47-s − 0.813·49-s + 0.285·51-s + 0.368·53-s + 0.0883·57-s − 1.28·59-s − 1.30·61-s + 0.367·63-s + 0.258·67-s + 0.261·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(129.804\)
Root analytic conductor: \(11.3931\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good3 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 + 25 T + p^{3} T^{2} \)
17 \( 1 - 52 T + p^{3} T^{2} \)
19 \( 1 - p T + p^{3} T^{2} \)
23 \( 1 - 75 T + p^{3} T^{2} \)
29 \( 1 - 241 T + p^{3} T^{2} \)
31 \( 1 - 189 T + p^{3} T^{2} \)
37 \( 1 + 68 T + p^{3} T^{2} \)
41 \( 1 - 240 T + p^{3} T^{2} \)
43 \( 1 - 183 T + p^{3} T^{2} \)
47 \( 1 - 56 T + p^{3} T^{2} \)
53 \( 1 - 142 T + p^{3} T^{2} \)
59 \( 1 + 584 T + p^{3} T^{2} \)
61 \( 1 + 622 T + p^{3} T^{2} \)
67 \( 1 - 142 T + p^{3} T^{2} \)
71 \( 1 - 435 T + p^{3} T^{2} \)
73 \( 1 + 628 T + p^{3} T^{2} \)
79 \( 1 + 824 T + p^{3} T^{2} \)
83 \( 1 + 837 T + p^{3} T^{2} \)
89 \( 1 + 53 T + p^{3} T^{2} \)
97 \( 1 + 1677 T + p^{3} T^{2} \)
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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

−8.280080310760753618490708495241, −7.68792054197539459256277257843, −6.76819503098643153551130787093, −5.94581736101449230733107553330, −5.15984303651024202734557734317, −4.23564549938027053426747394560, −2.96444377106785001424797397945, −2.74550728678563758014978305905, −1.20220762695352819136506799241, 0, 1.20220762695352819136506799241, 2.74550728678563758014978305905, 2.96444377106785001424797397945, 4.23564549938027053426747394560, 5.15984303651024202734557734317, 5.94581736101449230733107553330, 6.76819503098643153551130787093, 7.68792054197539459256277257843, 8.280080310760753618490708495241

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