Properties

Label 2-39e2-1.1-c3-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·2-s + 3.99·4-s − 13.8·5-s − 22.5·7-s + 13.8·8-s + 47.9·10-s + 22.5·11-s + 77.9·14-s − 80·16-s + 27·17-s − 88.3·19-s − 55.4·20-s − 77.9·22-s + 57·23-s + 66.9·25-s − 90.0·28-s + 69·29-s − 72.7·31-s + 166.·32-s − 93.5·34-s + 311.·35-s − 39.8·37-s + 306·38-s − 192.·40-s − 393.·41-s + 85·43-s + 90.0·44-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s − 1.23·5-s − 1.21·7-s + 0.612·8-s + 1.51·10-s + 0.617·11-s + 1.48·14-s − 1.25·16-s + 0.385·17-s − 1.06·19-s − 0.619·20-s − 0.755·22-s + 0.516·23-s + 0.535·25-s − 0.607·28-s + 0.441·29-s − 0.421·31-s + 0.918·32-s − 0.471·34-s + 1.50·35-s − 0.177·37-s + 1.30·38-s − 0.758·40-s − 1.49·41-s + 0.301·43-s + 0.308·44-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

\(L(2)\) \(\approx\) \(0.1829400896\)
\(L(\frac12)\) \(\approx\) \(0.1829400896\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.46T + 8T^{2} \)
5 \( 1 + 13.8T + 125T^{2} \)
7 \( 1 + 22.5T + 343T^{2} \)
11 \( 1 - 22.5T + 1.33e3T^{2} \)
17 \( 1 - 27T + 4.91e3T^{2} \)
19 \( 1 + 88.3T + 6.85e3T^{2} \)
23 \( 1 - 57T + 1.21e4T^{2} \)
29 \( 1 - 69T + 2.43e4T^{2} \)
31 \( 1 + 72.7T + 2.97e4T^{2} \)
37 \( 1 + 39.8T + 5.06e4T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 85T + 7.95e4T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 + 426T + 1.48e5T^{2} \)
59 \( 1 + 19.0T + 2.05e5T^{2} \)
61 \( 1 + 17T + 2.26e5T^{2} \)
67 \( 1 + 164.T + 3.00e5T^{2} \)
71 \( 1 + 583.T + 3.57e5T^{2} \)
73 \( 1 + 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 426.T + 5.71e5T^{2} \)
89 \( 1 + 306.T + 7.04e5T^{2} \)
97 \( 1 - 1.23e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.962254975805156522841103640230, −8.485118458944135114586963788419, −7.62468029585644588431638466048, −6.94479167643991053400292848981, −6.23500934074067587912894670163, −4.75200509128778970915759694154, −3.89151148736981725006837414788, −3.04593588660560850927418347817, −1.51608997029439906082986117727, −0.25975404239308756247164309666, 0.25975404239308756247164309666, 1.51608997029439906082986117727, 3.04593588660560850927418347817, 3.89151148736981725006837414788, 4.75200509128778970915759694154, 6.23500934074067587912894670163, 6.94479167643991053400292848981, 7.62468029585644588431638466048, 8.485118458944135114586963788419, 8.962254975805156522841103640230

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