Properties

Label 2-32487-1.1-c1-0-8
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-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 − 8·23-s + 3·24-s − 25-s + 26-s + 27-s − 6·29-s + 2·30-s + 8·31-s − 5·32-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 − 1.66·23-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 + 1.43·31-s − 0.883·32-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: \(1\)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.13391120157892, −14.90149806434437, −14.27807071094766, −13.65536449899501, −13.43972213510813, −12.56933198350827, −12.06425819404636, −11.65261097274765, −10.99140580344956, −10.27516392441236, −9.852950240710379, −9.369898431090872, −8.634241767125698, −8.429636231141922, −7.856607693604781, −7.311370453378898, −6.703361331226486, −6.029371079418115, −5.146170514380349, −4.279593592685560, −4.019301747726851, −3.607663926435165, −2.399606992831752, −1.802433492934002, −0.8548692494929604, 0, 0.8548692494929604, 1.802433492934002, 2.399606992831752, 3.607663926435165, 4.019301747726851, 4.279593592685560, 5.146170514380349, 6.029371079418115, 6.703361331226486, 7.311370453378898, 7.856607693604781, 8.429636231141922, 8.634241767125698, 9.369898431090872, 9.852950240710379, 10.27516392441236, 10.99140580344956, 11.65261097274765, 12.06425819404636, 12.56933198350827, 13.43972213510813, 13.65536449899501, 14.27807071094766, 14.90149806434437, 15.13391120157892

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