Properties

Label 2-32487-1.1-c1-0-15
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  + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s + 11-s − 2·12-s + 13-s − 2·15-s − 4·16-s + 17-s + 2·18-s − 2·19-s + 4·20-s + 2·22-s + 4·23-s − 25-s + 2·26-s − 27-s − 4·30-s − 5·31-s − 8·32-s − 33-s + 2·34-s + 2·36-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 + 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s + 0.894·20-s + 0.426·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.730·30-s − 0.898·31-s − 1.41·32-s − 0.174·33-s + 0.342·34-s + 1/3·36-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 \( 1 - p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.12699948624630, −14.73542120007616, −14.14058158081063, −13.63002858340630, −13.21777936745904, −12.82601021112162, −12.27779056185385, −11.65423007130614, −11.30826566037269, −10.65440041247929, −10.05407508797822, −9.487629566096687, −8.917862127854376, −8.260456340311266, −7.363211394971424, −6.597457357018208, −6.469128201720409, −5.686316731524634, −5.278815840761903, −4.837976814419278, −4.000577455120850, −3.550241006846177, −2.735467994651662, −1.999179460461132, −1.278678228768352, 0, 1.278678228768352, 1.999179460461132, 2.735467994651662, 3.550241006846177, 4.000577455120850, 4.837976814419278, 5.278815840761903, 5.686316731524634, 6.469128201720409, 6.597457357018208, 7.363211394971424, 8.260456340311266, 8.917862127854376, 9.487629566096687, 10.05407508797822, 10.65440041247929, 11.30826566037269, 11.65423007130614, 12.27779056185385, 12.82601021112162, 13.21777936745904, 13.63002858340630, 14.14058158081063, 14.73542120007616, 15.12699948624630

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