Properties

Label 2-32487-1.1-c1-0-12
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 3·5-s + 2·6-s + 9-s + 6·10-s − 6·11-s − 2·12-s + 13-s + 3·15-s − 4·16-s + 17-s − 2·18-s + 19-s − 6·20-s + 12·22-s − 5·23-s + 4·25-s − 2·26-s − 27-s − 29-s − 6·30-s − 31-s + 8·32-s + 6·33-s − 2·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s + 1/3·9-s + 1.89·10-s − 1.80·11-s − 0.577·12-s + 0.277·13-s + 0.774·15-s − 16-s + 0.242·17-s − 0.471·18-s + 0.229·19-s − 1.34·20-s + 2.55·22-s − 1.04·23-s + 4/5·25-s − 0.392·26-s − 0.192·27-s − 0.185·29-s − 1.09·30-s − 0.179·31-s + 1.41·32-s + 1.04·33-s − 0.342·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.84889360574452, −15.34969801087034, −14.81356790008228, −13.96562928484541, −13.21223139474747, −12.96373407108660, −12.03746686739496, −11.77895139440794, −11.15325751537467, −10.67787434344670, −10.27703159103685, −9.801029338079363, −9.019834367780571, −8.432424025840526, −7.925355981593068, −7.528675018339403, −7.294944526577764, −6.281852625874201, −5.734487478914867, −4.842197090028969, −4.461806330055660, −3.593841546693942, −2.846412935957008, −1.986162454255755, −1.080735801420912, 0, 0, 1.080735801420912, 1.986162454255755, 2.846412935957008, 3.593841546693942, 4.461806330055660, 4.842197090028969, 5.734487478914867, 6.281852625874201, 7.294944526577764, 7.528675018339403, 7.925355981593068, 8.432424025840526, 9.019834367780571, 9.801029338079363, 10.27703159103685, 10.67787434344670, 11.15325751537467, 11.77895139440794, 12.03746686739496, 12.96373407108660, 13.21223139474747, 13.96562928484541, 14.81356790008228, 15.34969801087034, 15.84889360574452

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