Properties

Label 2-2200-1.1-c3-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s − 13·7-s − 2·9-s − 11·11-s + 12·13-s + 67·17-s + 151·19-s + 65·21-s − 12·23-s + 145·27-s − 143·29-s − 337·31-s + 55·33-s + 125·37-s − 60·39-s − 240·41-s − 270·43-s − 448·47-s − 174·49-s − 335·51-s + 45·53-s − 755·57-s + 704·59-s − 217·61-s + 26·63-s − 284·67-s + 60·69-s + ⋯
L(s)  = 1  − 0.962·3-s − 0.701·7-s − 0.0740·9-s − 0.301·11-s + 0.256·13-s + 0.955·17-s + 1.82·19-s + 0.675·21-s − 0.108·23-s + 1.03·27-s − 0.915·29-s − 1.95·31-s + 0.290·33-s + 0.555·37-s − 0.246·39-s − 0.914·41-s − 0.957·43-s − 1.39·47-s − 0.507·49-s − 0.919·51-s + 0.116·53-s − 1.75·57-s + 1.55·59-s − 0.455·61-s + 0.0519·63-s − 0.517·67-s + 0.104·69-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: \(0\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8651260447\)
\(L(\frac12)\) \(\approx\) \(0.8651260447\)
\(L(\frac{5}{2})\) 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 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 + 13 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 67 T + p^{3} T^{2} \)
19 \( 1 - 151 T + p^{3} T^{2} \)
23 \( 1 + 12 T + p^{3} T^{2} \)
29 \( 1 + 143 T + p^{3} T^{2} \)
31 \( 1 + 337 T + p^{3} T^{2} \)
37 \( 1 - 125 T + p^{3} T^{2} \)
41 \( 1 + 240 T + p^{3} T^{2} \)
43 \( 1 + 270 T + p^{3} T^{2} \)
47 \( 1 + 448 T + p^{3} T^{2} \)
53 \( 1 - 45 T + p^{3} T^{2} \)
59 \( 1 - 704 T + p^{3} T^{2} \)
61 \( 1 + 217 T + p^{3} T^{2} \)
67 \( 1 + 284 T + p^{3} T^{2} \)
71 \( 1 + 515 T + p^{3} T^{2} \)
73 \( 1 - 1162 T + p^{3} T^{2} \)
79 \( 1 + 944 T + p^{3} T^{2} \)
83 \( 1 - 124 T + p^{3} T^{2} \)
89 \( 1 - 361 T + p^{3} T^{2} \)
97 \( 1 - 916 T + p^{3} T^{2} \)
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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

−8.759526577920169459533219666646, −7.77212726079119326018822344073, −7.09705171177404200989362944951, −6.22470393563774698673753020928, −5.47277189180418815043656197987, −5.09373076390742995409029242192, −3.63037604294184820166584047884, −3.07896796106935040064228253699, −1.58414524075762754126899226619, −0.44953160929816952137463727534, 0.44953160929816952137463727534, 1.58414524075762754126899226619, 3.07896796106935040064228253699, 3.63037604294184820166584047884, 5.09373076390742995409029242192, 5.47277189180418815043656197987, 6.22470393563774698673753020928, 7.09705171177404200989362944951, 7.77212726079119326018822344073, 8.759526577920169459533219666646

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