Properties

Label 2-39e2-1.1-c1-0-35
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  − 2.69·2-s + 5.24·4-s + 1.04·5-s + 0.554·7-s − 8.74·8-s − 2.82·10-s − 2.91·11-s − 1.49·14-s + 13.0·16-s + 4.85·17-s − 0.753·19-s + 5.50·20-s + 7.83·22-s − 5.76·23-s − 3.89·25-s + 2.91·28-s + 1.91·29-s − 9.51·31-s − 17.6·32-s − 13.0·34-s + 0.582·35-s + 5.75·37-s + 2.02·38-s − 9.16·40-s − 4.91·41-s − 11.0·43-s − 15.2·44-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s + 0.469·5-s + 0.209·7-s − 3.09·8-s − 0.892·10-s − 0.877·11-s − 0.399·14-s + 3.25·16-s + 1.17·17-s − 0.172·19-s + 1.23·20-s + 1.67·22-s − 1.20·23-s − 0.779·25-s + 0.550·28-s + 0.355·29-s − 1.70·31-s − 3.11·32-s − 2.23·34-s + 0.0983·35-s + 0.945·37-s + 0.328·38-s − 1.44·40-s − 0.767·41-s − 1.69·43-s − 2.30·44-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: $\chi_{1521} (1, \cdot )$
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 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 - 1.04T + 5T^{2} \)
7 \( 1 - 0.554T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 + 0.753T + 19T^{2} \)
23 \( 1 + 5.76T + 23T^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 + 9.51T + 31T^{2} \)
37 \( 1 - 5.75T + 37T^{2} \)
41 \( 1 + 4.91T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + 0.753T + 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 + 1.87T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 + 2.64T + 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−9.177755539424391430500654974412, −8.048507616854019405875601541002, −7.974127885516665872008237827783, −6.94357930852440145570557060532, −6.04570218867001902799632484874, −5.29259027200472378905888742564, −3.50442496680246461454648537459, −2.34558679142852012484468111324, −1.52304287975092734240630776917, 0, 1.52304287975092734240630776917, 2.34558679142852012484468111324, 3.50442496680246461454648537459, 5.29259027200472378905888742564, 6.04570218867001902799632484874, 6.94357930852440145570557060532, 7.974127885516665872008237827783, 8.048507616854019405875601541002, 9.177755539424391430500654974412

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