Properties

Label 2-32487-1.1-c1-0-11
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 + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s − 4·11-s − 12-s − 13-s + 2·15-s − 16-s − 17-s − 18-s + 4·19-s − 2·20-s + 4·22-s + 3·24-s − 25-s + 26-s + 27-s + 6·29-s − 2·30-s − 5·32-s − 4·33-s + 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.516·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.365·30-s − 0.883·32-s − 0.696·33-s + 0.171·34-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: $\chi_{32487} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.58958538878767, −14.55642282054943, −14.06821297054201, −13.86944574627533, −13.26889710379173, −12.79825323494695, −12.34508315647145, −11.42855211466595, −10.78017607433212, −10.24987852121666, −9.897186998237679, −9.383658786203668, −8.955033827211197, −8.250708236089537, −7.850384262366776, −7.339006318445019, −6.642767526082611, −5.777918444714075, −5.280026946003790, −4.690200434917036, −4.045358493863419, −3.073361872050615, −2.548229768256813, −1.781188984827314, −1.036476819782694, 0, 1.036476819782694, 1.781188984827314, 2.548229768256813, 3.073361872050615, 4.045358493863419, 4.690200434917036, 5.280026946003790, 5.777918444714075, 6.642767526082611, 7.339006318445019, 7.850384262366776, 8.250708236089537, 8.955033827211197, 9.383658786203668, 9.897186998237679, 10.24987852121666, 10.78017607433212, 11.42855211466595, 12.34508315647145, 12.79825323494695, 13.26889710379173, 13.86944574627533, 14.06821297054201, 14.55642282054943, 15.58958538878767

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