Properties

Label 2-32487-1.1-c1-0-5
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s − 2·11-s + 12-s + 13-s − 16-s − 17-s − 18-s + 2·22-s − 8·23-s − 3·24-s − 5·25-s − 26-s − 27-s − 6·29-s − 6·31-s − 5·32-s + 2·33-s + 34-s − 36-s + 4·37-s − 39-s + 12·41-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 − 0.603·11-s + 0.288·12-s + 0.277·13-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.426·22-s − 1.66·23-s − 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.883·32-s + 0.348·33-s + 0.171·34-s − 1/6·36-s + 0.657·37-s − 0.160·39-s + 1.87·41-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.54919374578202, −14.71395869803655, −14.25683847515861, −13.60707153330633, −13.21615307480334, −12.65675761933695, −12.18415644049152, −11.29864747058840, −11.07324401668395, −10.45353379404106, −9.850665530144944, −9.477260288807477, −8.961744391812376, −8.106577370337724, −7.796466131578672, −7.365477310021775, −6.421183103466588, −5.872431182239603, −5.355430213685897, −4.661441645687555, −3.986738823857492, −3.563027391399521, −2.242846755609036, −1.790511533655810, −0.7110840493957610, 0, 0.7110840493957610, 1.790511533655810, 2.242846755609036, 3.563027391399521, 3.986738823857492, 4.661441645687555, 5.355430213685897, 5.872431182239603, 6.421183103466588, 7.365477310021775, 7.796466131578672, 8.106577370337724, 8.961744391812376, 9.477260288807477, 9.850665530144944, 10.45353379404106, 11.07324401668395, 11.29864747058840, 12.18415644049152, 12.65675761933695, 13.21615307480334, 13.60707153330633, 14.25683847515861, 14.71395869803655, 15.54919374578202

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