Properties

Label 2-39e2-1.1-c1-0-34
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.24·2-s − 0.445·4-s + 2.80·5-s − 4.80·7-s + 3.04·8-s − 3.49·10-s − 1.46·11-s + 5.98·14-s − 2.91·16-s + 2.44·17-s + 2.54·19-s − 1.24·20-s + 1.82·22-s + 3.51·23-s + 2.85·25-s + 2.13·28-s − 1.85·29-s − 7.63·31-s − 2.46·32-s − 3.04·34-s − 13.4·35-s − 4.55·37-s − 3.17·38-s + 8.54·40-s − 1.24·41-s + 2.38·43-s + 0.652·44-s + ⋯
L(s)  = 1  − 0.881·2-s − 0.222·4-s + 1.25·5-s − 1.81·7-s + 1.07·8-s − 1.10·10-s − 0.442·11-s + 1.60·14-s − 0.727·16-s + 0.593·17-s + 0.583·19-s − 0.278·20-s + 0.389·22-s + 0.733·23-s + 0.570·25-s + 0.403·28-s − 0.343·29-s − 1.37·31-s − 0.436·32-s − 0.522·34-s − 2.27·35-s − 0.748·37-s − 0.514·38-s + 1.35·40-s − 0.194·41-s + 0.363·43-s + 0.0984·44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 1.24T + 2T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + 1.24T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 - 8.85T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 + 7.82T + 61T^{2} \)
67 \( 1 + 3.58T + 67T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 - 0.652T + 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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

−9.427112669523287395114987350414, −8.563137452600505583038366475641, −7.42971359446796852427066526973, −6.77676663227556136629680344727, −5.79726859550645205918331966796, −5.18318281863811362304794312606, −3.72021861383773522017958466972, −2.78142834641264025903654852269, −1.48025328363071115284434145052, 0, 1.48025328363071115284434145052, 2.78142834641264025903654852269, 3.72021861383773522017958466972, 5.18318281863811362304794312606, 5.79726859550645205918331966796, 6.77676663227556136629680344727, 7.42971359446796852427066526973, 8.563137452600505583038366475641, 9.427112669523287395114987350414

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