Properties

Label 2-2200-1.1-c1-0-28
Degree $2$
Conductor $2200$
Sign $1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·7-s + 6·9-s − 11-s + 6·17-s + 4·19-s + 6·21-s − 23-s + 9·27-s − 8·29-s − 7·31-s − 3·33-s + 37-s + 4·41-s − 6·43-s + 8·47-s − 3·49-s + 18·51-s − 2·53-s + 12·57-s − 59-s + 4·61-s + 12·63-s + 5·67-s − 3·69-s + 3·71-s − 16·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.755·7-s + 2·9-s − 0.301·11-s + 1.45·17-s + 0.917·19-s + 1.30·21-s − 0.208·23-s + 1.73·27-s − 1.48·29-s − 1.25·31-s − 0.522·33-s + 0.164·37-s + 0.624·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s + 2.52·51-s − 0.274·53-s + 1.58·57-s − 0.130·59-s + 0.512·61-s + 1.51·63-s + 0.610·67-s − 0.361·69-s + 0.356·71-s − 1.87·73-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.803578044\)
\(L(\frac12)\) \(\approx\) \(3.803578044\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−9.044429817831217181638998405705, −8.199474729672020673481485607421, −7.62229005238289566122645164588, −7.27337473671779027080168706120, −5.78186464770206751996559392839, −4.97733121051902579418088362815, −3.83371551925653981693071036397, −3.28552140533552989731814166401, −2.24522321731767850806358219473, −1.36958456151076610071741324322, 1.36958456151076610071741324322, 2.24522321731767850806358219473, 3.28552140533552989731814166401, 3.83371551925653981693071036397, 4.97733121051902579418088362815, 5.78186464770206751996559392839, 7.27337473671779027080168706120, 7.62229005238289566122645164588, 8.199474729672020673481485607421, 9.044429817831217181638998405705

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