Properties

Degree $2$
Conductor $441$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s − 3.41·5-s + 1.58·8-s + 1.41·10-s + 2·11-s + 2.58·13-s + 3·16-s + 2.24·17-s − 2.82·19-s + 6.24·20-s − 0.828·22-s + 7.65·23-s + 6.65·25-s − 1.07·26-s + 6.82·29-s − 1.17·31-s − 4.41·32-s − 0.928·34-s − 4·37-s + 1.17·38-s − 5.41·40-s − 6.24·41-s + 5.65·43-s − 3.65·44-s − 3.17·46-s + 2.82·47-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s − 1.52·5-s + 0.560·8-s + 0.447·10-s + 0.603·11-s + 0.717·13-s + 0.750·16-s + 0.543·17-s − 0.648·19-s + 1.39·20-s − 0.176·22-s + 1.59·23-s + 1.33·25-s − 0.210·26-s + 1.26·29-s − 0.210·31-s − 0.780·32-s − 0.159·34-s − 0.657·37-s + 0.190·38-s − 0.856·40-s − 0.974·41-s + 0.862·43-s − 0.551·44-s − 0.467·46-s + 0.412·47-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: $1$
Motivic weight: \(1\)
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.751214\)
\(L(\frac12)\) \(\approx\) \(0.751214\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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

−11.08816099052193730586026017416, −10.28656416713327785743945911650, −9.005376370676956116405339931168, −8.520270031082421588158231239740, −7.65630259062102721473457234019, −6.66328616045060353467405089350, −5.12967714732655062142470444546, −4.15787096548369957526349467029, −3.38373017834395905072088271255, −0.886873009659944755673861585793, 0.886873009659944755673861585793, 3.38373017834395905072088271255, 4.15787096548369957526349467029, 5.12967714732655062142470444546, 6.66328616045060353467405089350, 7.65630259062102721473457234019, 8.520270031082421588158231239740, 9.005376370676956116405339931168, 10.28656416713327785743945911650, 11.08816099052193730586026017416

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