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Properties

Label 2-32487-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 2·5-s + 2·6-s + 9-s + 4·10-s + 5·11-s + 2·12-s + 13-s + 2·15-s − 4·16-s + 17-s + 2·18-s + 6·19-s + 4·20-s + 10·22-s − 4·23-s − 25-s + 2·26-s + 27-s + 8·29-s + 4·30-s − 31-s − 8·32-s + 5·33-s + 2·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.894·5-s + 0.816·6-s + 1/3·9-s + 1.26·10-s + 1.50·11-s + 0.577·12-s + 0.277·13-s + 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s + 1.37·19-s + 0.894·20-s + 2.13·22-s − 0.834·23-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.48·29-s + 0.730·30-s − 0.179·31-s − 1.41·32-s + 0.870·33-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: \(0\)
Selberg data: \((2,\ 32487,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.920912547\)
\(L(\frac12)\) \(\approx\) \(9.920912547\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.61277083187154, −14.32677685512026, −14.01287570369509, −13.71358286727999, −13.06455515888382, −12.51879598968837, −11.99359320739600, −11.61462319591902, −10.98423888267335, −10.04248896189747, −9.739685777661591, −9.095945623613841, −8.737241929719872, −7.819919394303478, −7.229613559895115, −6.423339855366376, −6.188673163540017, −5.553549344207786, −4.938099016010476, −4.133347639968469, −3.839540426583355, −3.054633385005685, −2.510964423705192, −1.684273134306449, −0.9735986889105930, 0.9735986889105930, 1.684273134306449, 2.510964423705192, 3.054633385005685, 3.839540426583355, 4.133347639968469, 4.938099016010476, 5.553549344207786, 6.188673163540017, 6.423339855366376, 7.229613559895115, 7.819919394303478, 8.737241929719872, 9.095945623613841, 9.739685777661591, 10.04248896189747, 10.98423888267335, 11.61462319591902, 11.99359320739600, 12.51879598968837, 13.06455515888382, 13.71358286727999, 14.01287570369509, 14.32677685512026, 14.61277083187154

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