Properties

Label 2-44-1.1-c1-0-0
Degree $2$
Conductor $44$
Sign $1$
Analytic cond. $0.351341$
Root an. cond. $0.592740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 2·7-s − 2·9-s − 11-s − 4·13-s − 3·15-s + 6·17-s + 8·19-s + 2·21-s − 3·23-s + 4·25-s − 5·27-s + 5·31-s − 33-s − 6·35-s − 37-s − 4·39-s − 10·43-s + 6·45-s − 3·49-s + 6·51-s − 6·53-s + 3·55-s + 8·57-s + 3·59-s − 4·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s − 0.774·15-s + 1.45·17-s + 1.83·19-s + 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.962·27-s + 0.898·31-s − 0.174·33-s − 1.01·35-s − 0.164·37-s − 0.640·39-s − 1.52·43-s + 0.894·45-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.404·55-s + 1.05·57-s + 0.390·59-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8046462875\)
\(L(\frac12)\) \(\approx\) \(0.8046462875\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.77720419358441815590775762587, −14.73971481738632735611838632984, −13.98182431238434320022596333112, −12.11377477145794948883606144876, −11.52932662806962264853358891658, −9.813753565180213813561671136388, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −5.11227132195889762323685903220, −3.27381985666804460223319703873, 3.27381985666804460223319703873, 5.11227132195889762323685903220, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 9.813753565180213813561671136388, 11.52932662806962264853358891658, 12.11377477145794948883606144876, 13.98182431238434320022596333112, 14.73971481738632735611838632984, 15.77720419358441815590775762587

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