Properties

Label 2-39e2-1.1-c3-0-40
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s − 5·4-s − 1.73·5-s − 13.8·7-s + 22.5·8-s + 2.99·10-s + 13.8·11-s + 23.9·14-s + 1.00·16-s + 117·17-s + 114.·19-s + 8.66·20-s − 23.9·22-s − 78·23-s − 122·25-s + 69.2·28-s + 141·29-s − 155.·31-s − 181.·32-s − 202.·34-s + 23.9·35-s + 143.·37-s − 198·38-s − 38.9·40-s − 271.·41-s − 104·43-s − 69.2·44-s + ⋯
L(s)  = 1  − 0.612·2-s − 0.625·4-s − 0.154·5-s − 0.748·7-s + 0.995·8-s + 0.0948·10-s + 0.379·11-s + 0.458·14-s + 0.0156·16-s + 1.66·17-s + 1.38·19-s + 0.0968·20-s − 0.232·22-s − 0.707·23-s − 0.975·25-s + 0.467·28-s + 0.902·29-s − 0.903·31-s − 1.00·32-s − 1.02·34-s + 0.115·35-s + 0.638·37-s − 0.845·38-s − 0.154·40-s − 1.03·41-s − 0.368·43-s − 0.237·44-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: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.003731872\)
\(L(\frac12)\) \(\approx\) \(1.003731872\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.73T + 8T^{2} \)
5 \( 1 + 1.73T + 125T^{2} \)
7 \( 1 + 13.8T + 343T^{2} \)
11 \( 1 - 13.8T + 1.33e3T^{2} \)
17 \( 1 - 117T + 4.91e3T^{2} \)
19 \( 1 - 114.T + 6.85e3T^{2} \)
23 \( 1 + 78T + 1.21e4T^{2} \)
29 \( 1 - 141T + 2.43e4T^{2} \)
31 \( 1 + 155.T + 2.97e4T^{2} \)
37 \( 1 - 143.T + 5.06e4T^{2} \)
41 \( 1 + 271.T + 6.89e4T^{2} \)
43 \( 1 + 104T + 7.95e4T^{2} \)
47 \( 1 + 301.T + 1.03e5T^{2} \)
53 \( 1 + 93T + 1.48e5T^{2} \)
59 \( 1 - 284.T + 2.05e5T^{2} \)
61 \( 1 - 145T + 2.26e5T^{2} \)
67 \( 1 + 786.T + 3.00e5T^{2} \)
71 \( 1 - 1.05e3T + 3.57e5T^{2} \)
73 \( 1 - 458.T + 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 789.T + 5.71e5T^{2} \)
89 \( 1 + 976.T + 7.04e5T^{2} \)
97 \( 1 + 200.T + 9.12e5T^{2} \)
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

−9.327486932291249285724095862797, −8.206063465790956996325082526317, −7.77423134477657699827200500138, −6.83742857468351378204704595839, −5.77759699241434142784358207487, −5.02172039165984458568670852084, −3.82991968795437775432555786804, −3.22096572160771695183392359924, −1.57727971356795834644781448684, −0.57034227746350611523180720085, 0.57034227746350611523180720085, 1.57727971356795834644781448684, 3.22096572160771695183392359924, 3.82991968795437775432555786804, 5.02172039165984458568670852084, 5.77759699241434142784358207487, 6.83742857468351378204704595839, 7.77423134477657699827200500138, 8.206063465790956996325082526317, 9.327486932291249285724095862797

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