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Properties

Label 2-39e2-13.12-c1-0-16
Degree $2$
Conductor $1521$
Sign $-0.691 - 0.722i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.69i·2-s − 5.24·4-s − 1.04i·5-s + 0.554i·7-s − 8.74i·8-s + 2.82·10-s − 2.91i·11-s − 1.49·14-s + 13.0·16-s − 4.85·17-s + 0.753i·19-s + 5.50i·20-s + 7.83·22-s + 5.76·23-s + 3.89·25-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.62·4-s − 0.469i·5-s + 0.209i·7-s − 3.09i·8-s + 0.892·10-s − 0.877i·11-s − 0.399·14-s + 3.25·16-s − 1.17·17-s + 0.172i·19-s + 1.23i·20-s + 1.67·22-s + 1.20·23-s + 0.779·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.691 - 0.722i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.275052131\)
\(L(\frac12)\) \(\approx\) \(1.275052131\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.69iT - 2T^{2} \)
5 \( 1 + 1.04iT - 5T^{2} \)
7 \( 1 - 0.554iT - 7T^{2} \)
11 \( 1 + 2.91iT - 11T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 - 0.753iT - 19T^{2} \)
23 \( 1 - 5.76T + 23T^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 - 9.51iT - 31T^{2} \)
37 \( 1 - 5.75iT - 37T^{2} \)
41 \( 1 - 4.91iT - 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 0.753iT - 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 + 4.09iT - 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 - 1.87iT - 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 2.64iT - 83T^{2} \)
89 \( 1 + 9.92iT - 89T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−9.084858814569638535857610145611, −8.826128470742642894608940990459, −8.215208383590223271517718146511, −7.17521574983361334149751064976, −6.60687633628678451520635001283, −5.78445809390531759087797132160, −5.00171811512600499612870916933, −4.36965224806786746449139638394, −3.10880848788406165685559886560, −0.889525709410778723954731107381, 0.74513726900819630907797377474, 2.17297939242587534594821052711, 2.72400930474258247374171246681, 3.96162832640416538757463410137, 4.48569166050070118406325560932, 5.50338455864197124873915890832, 6.84569275732665781395397430848, 7.72980211126192189781484376346, 9.068499512857250588079924617721, 9.144916118001328978364477725475

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