Properties

Label 2-2200-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·3-s − 3.57·7-s + 4.57·9-s + 11-s − 13-s − 0.751·17-s − 2.50·19-s + 9.82·21-s − 5.75·23-s − 4.32·27-s − 4.07·29-s − 6.14·31-s − 2.75·33-s + 2.81·37-s + 2.75·39-s + 1.18·41-s − 7.68·43-s + 9.82·47-s + 5.75·49-s + 2.06·51-s − 14.2·53-s + 6.88·57-s − 9.82·59-s + 7.07·61-s − 16.3·63-s + 14.6·67-s + 15.8·69-s + ⋯
L(s)  = 1  − 1.58·3-s − 1.34·7-s + 1.52·9-s + 0.301·11-s − 0.277·13-s − 0.182·17-s − 0.574·19-s + 2.14·21-s − 1.19·23-s − 0.831·27-s − 0.756·29-s − 1.10·31-s − 0.478·33-s + 0.463·37-s + 0.440·39-s + 0.184·41-s − 1.17·43-s + 1.43·47-s + 0.821·49-s + 0.289·51-s − 1.95·53-s + 0.912·57-s − 1.27·59-s + 0.905·61-s − 2.05·63-s + 1.78·67-s + 1.90·69-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: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4240316841\)
\(L(\frac12)\) \(\approx\) \(0.4240316841\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.75T + 3T^{2} \)
7 \( 1 + 3.57T + 7T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 0.751T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + 5.75T + 23T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 - 2.81T + 37T^{2} \)
41 \( 1 - 1.18T + 41T^{2} \)
43 \( 1 + 7.68T + 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 9.82T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 3.89T + 79T^{2} \)
83 \( 1 - 5.57T + 83T^{2} \)
89 \( 1 - 7.42T + 89T^{2} \)
97 \( 1 - 0.609T + 97T^{2} \)
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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

−9.492891926342077188329936053187, −8.211459022331313316708857322891, −7.16299337478275751075407597677, −6.51617159201195940170550327891, −6.01683148790183519477888003266, −5.27619642434126653078725150849, −4.27721370510737966900077545707, −3.44993829169380887729889100490, −2.00810374247608213466951334310, −0.44086114792754937610326039081, 0.44086114792754937610326039081, 2.00810374247608213466951334310, 3.44993829169380887729889100490, 4.27721370510737966900077545707, 5.27619642434126653078725150849, 6.01683148790183519477888003266, 6.51617159201195940170550327891, 7.16299337478275751075407597677, 8.211459022331313316708857322891, 9.492891926342077188329936053187

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