Properties

Label 2-39e2-1.1-c3-0-114
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.149·2-s − 7.97·4-s + 10.2·5-s − 29.6·7-s + 2.38·8-s − 1.52·10-s − 38.1·11-s + 4.42·14-s + 63.4·16-s + 71.3·17-s + 10.0·19-s − 81.7·20-s + 5.67·22-s + 198.·23-s − 19.8·25-s + 236.·28-s − 30.8·29-s + 151.·31-s − 28.5·32-s − 10.6·34-s − 304.·35-s + 151.·37-s − 1.50·38-s + 24.4·40-s + 207.·41-s − 303.·43-s + 303.·44-s + ⋯
L(s)  = 1  − 0.0527·2-s − 0.997·4-s + 0.917·5-s − 1.60·7-s + 0.105·8-s − 0.0483·10-s − 1.04·11-s + 0.0844·14-s + 0.991·16-s + 1.01·17-s + 0.121·19-s − 0.914·20-s + 0.0550·22-s + 1.80·23-s − 0.159·25-s + 1.59·28-s − 0.197·29-s + 0.878·31-s − 0.157·32-s − 0.0536·34-s − 1.46·35-s + 0.672·37-s − 0.00642·38-s + 0.0965·40-s + 0.789·41-s − 1.07·43-s + 1.04·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 0.149T + 8T^{2} \)
5 \( 1 - 10.2T + 125T^{2} \)
7 \( 1 + 29.6T + 343T^{2} \)
11 \( 1 + 38.1T + 1.33e3T^{2} \)
17 \( 1 - 71.3T + 4.91e3T^{2} \)
19 \( 1 - 10.0T + 6.85e3T^{2} \)
23 \( 1 - 198.T + 1.21e4T^{2} \)
29 \( 1 + 30.8T + 2.43e4T^{2} \)
31 \( 1 - 151.T + 2.97e4T^{2} \)
37 \( 1 - 151.T + 5.06e4T^{2} \)
41 \( 1 - 207.T + 6.89e4T^{2} \)
43 \( 1 + 303.T + 7.95e4T^{2} \)
47 \( 1 + 12.2T + 1.03e5T^{2} \)
53 \( 1 - 250.T + 1.48e5T^{2} \)
59 \( 1 + 390.T + 2.05e5T^{2} \)
61 \( 1 + 156.T + 2.26e5T^{2} \)
67 \( 1 + 303.T + 3.00e5T^{2} \)
71 \( 1 + 913.T + 3.57e5T^{2} \)
73 \( 1 + 249.T + 3.89e5T^{2} \)
79 \( 1 + 147.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 946.T + 7.04e5T^{2} \)
97 \( 1 - 417.T + 9.12e5T^{2} \)
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

−8.987295072472112574390580795821, −7.965060903011138460418776588256, −7.07596398556783298779729256684, −6.02543284864781601538394380326, −5.52096876913378670819267487684, −4.59388329345945382010881070955, −3.32851572485504237770231940900, −2.75392345997472372101979916403, −1.10368376252315050743894441348, 0, 1.10368376252315050743894441348, 2.75392345997472372101979916403, 3.32851572485504237770231940900, 4.59388329345945382010881070955, 5.52096876913378670819267487684, 6.02543284864781601538394380326, 7.07596398556783298779729256684, 7.965060903011138460418776588256, 8.987295072472112574390580795821

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