Properties

Degree $2$
Conductor $32487$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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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 − 2·29-s − 2·30-s + 8·31-s − 5·32-s + 4·33-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 − 0.371·29-s − 0.365·30-s + 1.43·31-s − 0.883·32-s + 0.696·33-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{32487} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.25360273982723, −14.65282980466102, −14.09530035650785, −13.77948737220147, −13.44616209216974, −12.76973394407322, −12.12316405733530, −11.57921797018903, −10.88832180043411, −10.15632217817647, −9.822040785289717, −9.307172869672124, −9.077268392204329, −8.328695316978420, −7.848461449020156, −7.257348821742786, −6.474444399202381, −6.119738642247438, −5.055460654250835, −4.773111017113988, −3.902334102268935, −3.325573810103288, −2.458935197952399, −1.538706369946090, −1.281931186044459, 0, 1.281931186044459, 1.538706369946090, 2.458935197952399, 3.325573810103288, 3.902334102268935, 4.773111017113988, 5.055460654250835, 6.119738642247438, 6.474444399202381, 7.257348821742786, 7.848461449020156, 8.328695316978420, 9.077268392204329, 9.307172869672124, 9.822040785289717, 10.15632217817647, 10.88832180043411, 11.57921797018903, 12.12316405733530, 12.76973394407322, 13.44616209216974, 13.77948737220147, 14.09530035650785, 14.65282980466102, 15.25360273982723

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