Properties

Label 2-39e2-1.1-c1-0-39
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  − 2.45·2-s + 4.04·4-s + 3.33·5-s − 3.69·7-s − 5.03·8-s − 8.20·10-s − 0.270·11-s + 9.08·14-s + 4.29·16-s + 4.04·17-s − 7.15·19-s + 13.5·20-s + 0.664·22-s + 2.79·23-s + 6.13·25-s − 14.9·28-s − 7.83·29-s − 2.14·31-s − 0.487·32-s − 9.93·34-s − 12.3·35-s − 4.37·37-s + 17.6·38-s − 16.8·40-s − 7.61·41-s − 3.89·43-s − 1.09·44-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.02·4-s + 1.49·5-s − 1.39·7-s − 1.78·8-s − 2.59·10-s − 0.0815·11-s + 2.42·14-s + 1.07·16-s + 0.980·17-s − 1.64·19-s + 3.02·20-s + 0.141·22-s + 0.583·23-s + 1.22·25-s − 2.82·28-s − 1.45·29-s − 0.385·31-s − 0.0861·32-s − 1.70·34-s − 2.08·35-s − 0.719·37-s + 2.85·38-s − 2.65·40-s − 1.18·41-s − 0.593·43-s − 0.165·44-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 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 - 3.33T + 5T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 + 0.270T + 11T^{2} \)
17 \( 1 - 4.04T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 - 2.79T + 23T^{2} \)
29 \( 1 + 7.83T + 29T^{2} \)
31 \( 1 + 2.14T + 31T^{2} \)
37 \( 1 + 4.37T + 37T^{2} \)
41 \( 1 + 7.61T + 41T^{2} \)
43 \( 1 + 3.89T + 43T^{2} \)
47 \( 1 - 0.877T + 47T^{2} \)
53 \( 1 + 6.83T + 53T^{2} \)
59 \( 1 - 3.40T + 59T^{2} \)
61 \( 1 - 3.06T + 61T^{2} \)
67 \( 1 - 5.00T + 67T^{2} \)
71 \( 1 + 7.59T + 71T^{2} \)
73 \( 1 - 0.405T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 6.98T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.167770772784264880493039349812, −8.646802485176583542872214698596, −7.54406111260861141897583092343, −6.65147435542939920414151934418, −6.24417363144239276125351478263, −5.31194935683536856353881318696, −3.47400178013755761644798392882, −2.40827694742778115771991720295, −1.55671476947290960686366092336, 0, 1.55671476947290960686366092336, 2.40827694742778115771991720295, 3.47400178013755761644798392882, 5.31194935683536856353881318696, 6.24417363144239276125351478263, 6.65147435542939920414151934418, 7.54406111260861141897583092343, 8.646802485176583542872214698596, 9.167770772784264880493039349812

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