Properties

Label 2-42e2-1.1-c1-0-0
Degree $2$
Conductor $1764$
Sign $1$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.74·5-s − 5.29·11-s + 4.24·13-s − 3.74·17-s − 2.82·19-s + 5.29·23-s + 9·25-s + 5.29·29-s − 8.48·31-s + 4·37-s − 3.74·41-s + 8·43-s + 7.48·47-s + 10.5·53-s + 19.7·55-s + 7.48·59-s − 9.89·61-s − 15.8·65-s + 12·67-s − 15.8·71-s − 1.41·73-s − 4·79-s + 14.9·83-s + 14·85-s + 3.74·89-s + 10.5·95-s + 9.89·97-s + ⋯
L(s)  = 1  − 1.67·5-s − 1.59·11-s + 1.17·13-s − 0.907·17-s − 0.648·19-s + 1.10·23-s + 1.80·25-s + 0.982·29-s − 1.52·31-s + 0.657·37-s − 0.584·41-s + 1.21·43-s + 1.09·47-s + 1.45·53-s + 2.66·55-s + 0.974·59-s − 1.26·61-s − 1.96·65-s + 1.46·67-s − 1.88·71-s − 0.165·73-s − 0.450·79-s + 1.64·83-s + 1.51·85-s + 0.396·89-s + 1.08·95-s + 1.00·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9024826411\)
\(L(\frac12)\) \(\approx\) \(0.9024826411\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.74T + 5T^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 7.48T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 7.48T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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

−8.843813299794426708421962779897, −8.616066901731153798918336187514, −7.65300516974763175449139132806, −7.20119746762538833727372999184, −6.12258684548066016523315839710, −5.05372802939361225052039845336, −4.25950782223466240652777476122, −3.45997770503630707529680616475, −2.45999054854238803729820927177, −0.63021203078849393912758571123, 0.63021203078849393912758571123, 2.45999054854238803729820927177, 3.45997770503630707529680616475, 4.25950782223466240652777476122, 5.05372802939361225052039845336, 6.12258684548066016523315839710, 7.20119746762538833727372999184, 7.65300516974763175449139132806, 8.616066901731153798918336187514, 8.843813299794426708421962779897

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