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Properties

Label 2-32487-1.1-c1-0-16
Degree $2$
Conductor $32487$
Sign $-1$
Analytic cond. $259.410$
Root an. cond. $16.1062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 9-s − 2·10-s + 2·11-s + 2·12-s − 13-s − 15-s − 4·16-s − 17-s + 2·18-s + 7·19-s − 2·20-s + 4·22-s − 9·23-s − 4·25-s − 2·26-s + 27-s + 3·29-s − 2·30-s − 3·31-s − 8·32-s + 2·33-s − 2·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.277·13-s − 0.258·15-s − 16-s − 0.242·17-s + 0.471·18-s + 1.60·19-s − 0.447·20-s + 0.852·22-s − 1.87·23-s − 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.557·29-s − 0.365·30-s − 0.538·31-s − 1.41·32-s + 0.348·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32487\)    =    \(3 \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(259.410\)
Root analytic conductor: \(16.1062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32487,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.01153499221865, −14.65619003777378, −14.21015160906822, −13.80128122002895, −13.33414529042622, −12.76889878816532, −12.10898108768541, −11.77538696832965, −11.49042472320857, −10.59476406345872, −9.868359292124485, −9.435597453327073, −8.865640503983461, −8.017096735633975, −7.708091633158551, −6.904399562058139, −6.483106466391201, −5.527690842354169, −5.434538235427940, −4.270506538206435, −4.111205876761376, −3.534043100444259, −2.796065352426352, −2.214289509273674, −1.280584055195388, 0, 1.280584055195388, 2.214289509273674, 2.796065352426352, 3.534043100444259, 4.111205876761376, 4.270506538206435, 5.434538235427940, 5.527690842354169, 6.483106466391201, 6.904399562058139, 7.708091633158551, 8.017096735633975, 8.865640503983461, 9.435597453327073, 9.868359292124485, 10.59476406345872, 11.49042472320857, 11.77538696832965, 12.10898108768541, 12.76889878816532, 13.33414529042622, 13.80128122002895, 14.21015160906822, 14.65619003777378, 15.01153499221865

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