Properties

Label 2-39e2-1.1-c1-0-41
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 0.999·4-s + 1.73·5-s + 1.73·8-s − 2.99·10-s − 5·16-s − 3·17-s − 3.46·19-s + 1.73·20-s − 6·23-s − 2.00·25-s − 3·29-s + 3.46·31-s + 5.19·32-s + 5.19·34-s + 8.66·37-s + 5.99·38-s + 3.00·40-s − 5.19·41-s − 8·43-s + 10.3·46-s − 3.46·47-s − 7·49-s + 3.46·50-s + 3·53-s + 5.19·58-s + 6.92·59-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s + 0.774·5-s + 0.612·8-s − 0.948·10-s − 1.25·16-s − 0.727·17-s − 0.794·19-s + 0.387·20-s − 1.25·23-s − 0.400·25-s − 0.557·29-s + 0.622·31-s + 0.918·32-s + 0.891·34-s + 1.42·37-s + 0.973·38-s + 0.474·40-s − 0.811·41-s − 1.21·43-s + 1.53·46-s − 0.505·47-s − 49-s + 0.489·50-s + 0.412·53-s + 0.682·58-s + 0.901·59-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 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 8.66T + 37T^{2} \)
41 \( 1 + 5.19T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 1.73T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 - 6.92T + 97T^{2} \)
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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

−9.090624732530601026310937141632, −8.387822619347924294499169090191, −7.75884460601616925072064410954, −6.70435361669459934445453654506, −6.06398394553867898190904869553, −4.90381927800315230984413086383, −3.97438009968865693535853776413, −2.39072565639573916612797932788, −1.59421684282812097897887678615, 0, 1.59421684282812097897887678615, 2.39072565639573916612797932788, 3.97438009968865693535853776413, 4.90381927800315230984413086383, 6.06398394553867898190904869553, 6.70435361669459934445453654506, 7.75884460601616925072064410954, 8.387822619347924294499169090191, 9.090624732530601026310937141632

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