Properties

Label 2-39e2-1.1-c3-0-162
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.42·2-s + 11.6·4-s − 20.3·5-s + 15.9·8-s − 90.2·10-s + 70.0·11-s − 22.1·16-s − 236.·20-s + 310.·22-s + 290.·25-s − 225.·32-s − 325.·40-s − 486.·41-s − 452·43-s + 812.·44-s − 71.1·47-s − 343·49-s + 1.28e3·50-s − 1.42e3·55-s − 696.·59-s − 944.·61-s − 822.·64-s − 123.·71-s + 418.·79-s + 451.·80-s − 2.15e3·82-s + 1.50e3·83-s + ⋯
L(s)  = 1  + 1.56·2-s + 1.45·4-s − 1.82·5-s + 0.705·8-s − 2.85·10-s + 1.91·11-s − 0.346·16-s − 2.64·20-s + 3.00·22-s + 2.32·25-s − 1.24·32-s − 1.28·40-s − 1.85·41-s − 1.60·43-s + 2.78·44-s − 0.220·47-s − 49-s + 3.64·50-s − 3.50·55-s − 1.53·59-s − 1.98·61-s − 1.60·64-s − 0.206·71-s + 0.595·79-s + 0.631·80-s − 2.90·82-s + 1.99·83-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 \( 1 - 4.42T + 8T^{2} \)
5 \( 1 + 20.3T + 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 70.0T + 1.33e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 5.06e4T^{2} \)
41 \( 1 + 486.T + 6.89e4T^{2} \)
43 \( 1 + 452T + 7.95e4T^{2} \)
47 \( 1 + 71.1T + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 696.T + 2.05e5T^{2} \)
61 \( 1 + 944.T + 2.26e5T^{2} \)
67 \( 1 + 3.00e5T^{2} \)
71 \( 1 + 123.T + 3.57e5T^{2} \)
73 \( 1 + 3.89e5T^{2} \)
79 \( 1 - 418.T + 4.93e5T^{2} \)
83 \( 1 - 1.50e3T + 5.71e5T^{2} \)
89 \( 1 - 155.T + 7.04e5T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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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

−8.572961381240159222404037465508, −7.68261532937547794948260948361, −6.74959183188466457389478152586, −6.37291978815668365772005135821, −4.97316708748453382701676845352, −4.41636826877582856312733702961, −3.59149591061077614802528530241, −3.24681318391671947977094535206, −1.53515660561355164867993949043, 0, 1.53515660561355164867993949043, 3.24681318391671947977094535206, 3.59149591061077614802528530241, 4.41636826877582856312733702961, 4.97316708748453382701676845352, 6.37291978815668365772005135821, 6.74959183188466457389478152586, 7.68261532937547794948260948361, 8.572961381240159222404037465508

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