Properties

Label 2-2200-88.21-c0-0-2
Degree $2$
Conductor $2200$
Sign $1$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 9-s − 11-s + 13-s + 16-s − 18-s + 19-s + 22-s − 23-s − 26-s + 29-s − 31-s − 32-s + 36-s − 38-s + 43-s − 44-s + 46-s + 2·47-s + 49-s + 52-s − 58-s − 2·61-s + 62-s + 64-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s − 11-s + 13-s + 16-s − 18-s + 19-s + 22-s − 23-s − 26-s + 29-s − 31-s − 32-s + 36-s − 38-s + 43-s − 44-s + 46-s + 2·47-s + 49-s + 52-s − 58-s − 2·61-s + 62-s + 64-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2200} (901, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8299727889\)
\(L(\frac12)\) \(\approx\) \(0.8299727889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 - T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.207600487062802002944705460007, −8.535919785732311325773619159058, −7.57692047989688062630002603538, −7.33472121979338669320753548578, −6.18307758978877698735232808703, −5.56303361328060800891733320419, −4.30773807418554101799682799388, −3.27524622575987675270675734156, −2.19949185063119784717434905858, −1.07927288742098398680233742823, 1.07927288742098398680233742823, 2.19949185063119784717434905858, 3.27524622575987675270675734156, 4.30773807418554101799682799388, 5.56303361328060800891733320419, 6.18307758978877698735232808703, 7.33472121979338669320753548578, 7.57692047989688062630002603538, 8.535919785732311325773619159058, 9.207600487062802002944705460007

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