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Properties

Label 2-32487-1.1-c1-0-10
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-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s − 13-s − 16-s + 17-s + 18-s + 8·19-s + 4·22-s + 3·24-s − 5·25-s − 26-s − 27-s + 8·29-s + 8·31-s + 5·32-s − 4·33-s + 34-s − 36-s − 12·37-s + 8·38-s + 39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.852·22-s + 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.883·32-s − 0.696·33-s + 0.171·34-s − 1/6·36-s − 1.97·37-s + 1.29·38-s + 0.160·39-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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.36702565886421, −14.61581775962330, −14.07112646784507, −13.69779623039457, −13.50388382984969, −12.41508239655022, −12.09585649079779, −11.87406929280133, −11.37754959547973, −10.38212119155456, −9.924600528818172, −9.540480743060871, −8.858067863970960, −8.294054404405608, −7.634825783448357, −6.754785420659470, −6.500832598742178, −5.758616430855105, −5.143681007956233, −4.772726786332370, −4.091691718600062, −3.350320038957422, −3.014811474819043, −1.686205545295087, −1.028486289475035, 0, 1.028486289475035, 1.686205545295087, 3.014811474819043, 3.350320038957422, 4.091691718600062, 4.772726786332370, 5.143681007956233, 5.758616430855105, 6.500832598742178, 6.754785420659470, 7.634825783448357, 8.294054404405608, 8.858067863970960, 9.540480743060871, 9.924600528818172, 10.38212119155456, 11.37754959547973, 11.87406929280133, 12.09585649079779, 12.41508239655022, 13.50388382984969, 13.69779623039457, 14.07112646784507, 14.61581775962330, 15.36702565886421

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