Properties

Label 2-21e2-1.1-c3-0-29
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s + 45·8-s + 68·11-s + 89·16-s + 340·22-s + 40·23-s − 125·25-s + 166·29-s + 85·32-s + 450·37-s − 180·43-s + 1.15e3·44-s + 200·46-s − 625·50-s − 590·53-s + 830·58-s − 287·64-s − 740·67-s − 688·71-s + 2.25e3·74-s − 1.38e3·79-s − 900·86-s + 3.06e3·88-s + 680·92-s − 2.12e3·100-s − 2.95e3·106-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s + 1.98·8-s + 1.86·11-s + 1.39·16-s + 3.29·22-s + 0.362·23-s − 25-s + 1.06·29-s + 0.469·32-s + 1.99·37-s − 0.638·43-s + 3.96·44-s + 0.641·46-s − 1.76·50-s − 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s − 1.15·71-s + 3.53·74-s − 1.97·79-s − 1.12·86-s + 3.70·88-s + 0.770·92-s − 2.12·100-s − 2.70·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.197291470\)
\(L(\frac12)\) \(\approx\) \(6.197291470\)
\(L(\frac{5}{2})\) 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 \( 1 - 5 T + p^{3} T^{2} \)
5 \( 1 + p^{3} T^{2} \)
11 \( 1 - 68 T + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 - 40 T + p^{3} T^{2} \)
29 \( 1 - 166 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 - 450 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 180 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + 590 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + p^{3} T^{2} \)
67 \( 1 + 740 T + p^{3} T^{2} \)
71 \( 1 + 688 T + p^{3} T^{2} \)
73 \( 1 + p^{3} T^{2} \)
79 \( 1 + 1384 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + p^{3} T^{2} \)
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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

−11.31435525835004719778360066343, −9.983927217498222619537261635393, −8.919698787967461541489595300935, −7.54254320009823258691593577599, −6.49740437123140591846158446986, −5.99316120484815613506263511199, −4.67113003251447696092695034138, −3.98515496184856310144687655078, −2.92446605741448292310787102731, −1.48976179735318305341387801246, 1.48976179735318305341387801246, 2.92446605741448292310787102731, 3.98515496184856310144687655078, 4.67113003251447696092695034138, 5.99316120484815613506263511199, 6.49740437123140591846158446986, 7.54254320009823258691593577599, 8.919698787967461541489595300935, 9.983927217498222619537261635393, 11.31435525835004719778360066343

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