Properties

Label 2-39e2-1.1-c3-0-1
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  − 3.85·2-s + 6.86·4-s − 19.5·5-s + 6.26·7-s + 4.36·8-s + 75.3·10-s − 27.2·11-s − 24.1·14-s − 71.7·16-s − 30.8·17-s − 127.·19-s − 134.·20-s + 105.·22-s − 84.3·23-s + 256.·25-s + 43.0·28-s − 272.·29-s + 166.·31-s + 241.·32-s + 118.·34-s − 122.·35-s + 198.·37-s + 492.·38-s − 85.2·40-s + 160.·41-s + 158.·43-s − 187.·44-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.858·4-s − 1.74·5-s + 0.338·7-s + 0.192·8-s + 2.38·10-s − 0.746·11-s − 0.461·14-s − 1.12·16-s − 0.440·17-s − 1.54·19-s − 1.49·20-s + 1.01·22-s − 0.764·23-s + 2.05·25-s + 0.290·28-s − 1.74·29-s + 0.963·31-s + 1.33·32-s + 0.599·34-s − 0.590·35-s + 0.880·37-s + 2.10·38-s − 0.337·40-s + 0.610·41-s + 0.563·43-s − 0.641·44-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

\(L(2)\) \(\approx\) \(0.02099163949\)
\(L(\frac12)\) \(\approx\) \(0.02099163949\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.85T + 8T^{2} \)
5 \( 1 + 19.5T + 125T^{2} \)
7 \( 1 - 6.26T + 343T^{2} \)
11 \( 1 + 27.2T + 1.33e3T^{2} \)
17 \( 1 + 30.8T + 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 + 84.3T + 1.21e4T^{2} \)
29 \( 1 + 272.T + 2.43e4T^{2} \)
31 \( 1 - 166.T + 2.97e4T^{2} \)
37 \( 1 - 198.T + 5.06e4T^{2} \)
41 \( 1 - 160.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 + 305.T + 1.03e5T^{2} \)
53 \( 1 + 356.T + 1.48e5T^{2} \)
59 \( 1 + 470.T + 2.05e5T^{2} \)
61 \( 1 + 171.T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3T + 3.00e5T^{2} \)
71 \( 1 + 188.T + 3.57e5T^{2} \)
73 \( 1 + 959.T + 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 105.T + 5.71e5T^{2} \)
89 \( 1 + 649.T + 7.04e5T^{2} \)
97 \( 1 + 707.T + 9.12e5T^{2} \)
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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

−8.874608269715025146524522039667, −8.221207891276316775000035293339, −7.79889253671292996159653864469, −7.19483429527654577707034648683, −6.11741343174577976432542809306, −4.57908819378458664304511674055, −4.19349295601909716129310325048, −2.81550082141436203728273721138, −1.58434371322442796836646098508, −0.087232641167494068749676782804, 0.087232641167494068749676782804, 1.58434371322442796836646098508, 2.81550082141436203728273721138, 4.19349295601909716129310325048, 4.57908819378458664304511674055, 6.11741343174577976432542809306, 7.19483429527654577707034648683, 7.79889253671292996159653864469, 8.221207891276316775000035293339, 8.874608269715025146524522039667

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