↑

Properties

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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s + 9-s + 6·11-s − 2·12-s + 13-s − 15-s + 4·16-s − 17-s − 7·19-s + 2·20-s − 3·23-s − 4·25-s + 27-s − 7·29-s − 7·31-s + 6·33-s − 2·36-s + 8·37-s + 39-s − 2·41-s − 9·43-s − 12·44-s − 45-s − 7·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s − 0.242·17-s − 1.60·19-s + 0.447·20-s − 0.625·23-s − 4/5·25-s + 0.192·27-s − 1.29·29-s − 1.25·31-s + 1.04·33-s − 1/3·36-s + 1.31·37-s + 0.160·39-s − 0.312·41-s − 1.37·43-s − 1.80·44-s − 0.149·45-s − 1.02·47-s + 0.577·48-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

\(L(1)\) \(\approx\) \(1.312715451\)
\(L(\frac12)\) \(\approx\) \(1.312715451\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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

−14.92206251538509, −14.54978533242510, −14.07049921361483, −13.49108926857741, −12.86758266197280, −12.69047252199211, −11.78902839157666, −11.44068880681403, −10.79758296917969, −10.01754719751421, −9.480814621135192, −9.141745559810285, −8.581379959638945, −8.098571660816901, −7.594384870344474, −6.698863244184278, −6.309792091182054, −5.583624882106934, −4.713820937076669, −4.066592950464651, −3.853126817021179, −3.297809181085232, −2.013056746422796, −1.584166516790419, −0.4244782984772047, 0.4244782984772047, 1.584166516790419, 2.013056746422796, 3.297809181085232, 3.853126817021179, 4.066592950464651, 4.713820937076669, 5.583624882106934, 6.309792091182054, 6.698863244184278, 7.594384870344474, 8.098571660816901, 8.581379959638945, 9.141745559810285, 9.480814621135192, 10.01754719751421, 10.79758296917969, 11.44068880681403, 11.78902839157666, 12.69047252199211, 12.86758266197280, 13.49108926857741, 14.07049921361483, 14.54978533242510, 14.92206251538509

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