Properties

Label 2-21e2-1.1-c1-0-3
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7·13-s + 4·16-s + 7·19-s − 5·25-s + 7·31-s − 37-s + 5·43-s − 14·52-s − 14·61-s − 8·64-s + 11·67-s + 7·73-s − 14·76-s − 13·79-s − 14·97-s + 10·100-s + 7·103-s + 17·109-s + ⋯
L(s)  = 1  − 4-s + 1.94·13-s + 16-s + 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s + 0.762·43-s − 1.94·52-s − 1.79·61-s − 64-s + 1.34·67-s + 0.819·73-s − 1.60·76-s − 1.46·79-s − 1.42·97-s + 100-s + 0.689·103-s + 1.62·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209177364\)
\(L(\frac12)\) \(\approx\) \(1.209177364\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−11.12611090306175738214534717990, −10.09209954751813487013591320496, −9.291988044548486732019479222726, −8.461373500657640359204806994178, −7.67458414128501773545216406647, −6.24141795466275696941914757951, −5.40791154265456502263047742620, −4.19234808180063134723072414533, −3.28050491254919886424015350098, −1.14668647924786773157165026874, 1.14668647924786773157165026874, 3.28050491254919886424015350098, 4.19234808180063134723072414533, 5.40791154265456502263047742620, 6.24141795466275696941914757951, 7.67458414128501773545216406647, 8.461373500657640359204806994178, 9.291988044548486732019479222726, 10.09209954751813487013591320496, 11.12611090306175738214534717990

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