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Properties

Label 2-1100-11.10-c0-0-1
Degree $2$
Conductor $1100$
Sign $1$
Analytic cond. $0.548971$
Root an. cond. $0.740926$
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  + 3-s + 11-s + 23-s − 27-s − 31-s + 33-s + 37-s − 2·47-s + 49-s − 2·53-s − 59-s + 67-s + 69-s − 71-s − 81-s − 89-s − 93-s + 97-s − 2·103-s + 111-s + 113-s + ⋯
L(s)  = 1  + 3-s + 11-s + 23-s − 27-s − 31-s + 33-s + 37-s − 2·47-s + 49-s − 2·53-s − 59-s + 67-s + 69-s − 71-s − 81-s − 89-s − 93-s + 97-s − 2·103-s + 111-s + 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1100} (901, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432648494\)
\(L(\frac12)\) \(\approx\) \(1.432648494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.694491748062756641011246352633, −9.234998055604318450713514121998, −8.495821575895335710433655452420, −7.71196370436021627551023001648, −6.82044501060196204280740388653, −5.89671648638049700732012787382, −4.72288648487013544354478803953, −3.67217217624074011508594178346, −2.88161968926659634223196273119, −1.62500272502175647556573203193, 1.62500272502175647556573203193, 2.88161968926659634223196273119, 3.67217217624074011508594178346, 4.72288648487013544354478803953, 5.89671648638049700732012787382, 6.82044501060196204280740388653, 7.71196370436021627551023001648, 8.495821575895335710433655452420, 9.234998055604318450713514121998, 9.694491748062756641011246352633

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