Properties

Label 2-2200-88.21-c0-0-8
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

\(L(\frac{1}{2})\) \(\approx\) \(2.303338364\)
\(L(\frac12)\) \(\approx\) \(2.303338364\)
\(L(1)\) 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

−9.451090910271737869441272200118, −8.210346721278915778772859851097, −7.34067576540953108527614575660, −7.06453802486534580842693970504, −5.96171503832860490509432234987, −4.96963274052946164368422145510, −4.71449100694708243412791997382, −3.43904259109704609897937225722, −2.68787756596712922040962031221, −1.52917852974905540412251339942, 1.52917852974905540412251339942, 2.68787756596712922040962031221, 3.43904259109704609897937225722, 4.71449100694708243412791997382, 4.96963274052946164368422145510, 5.96171503832860490509432234987, 7.06453802486534580842693970504, 7.34067576540953108527614575660, 8.210346721278915778772859851097, 9.451090910271737869441272200118

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