Properties

Label 2-32487-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 4·5-s + 6-s + 3·8-s + 9-s − 4·10-s − 4·11-s + 12-s − 13-s − 4·15-s − 16-s − 17-s − 18-s − 4·20-s + 4·22-s − 4·23-s − 3·24-s + 11·25-s + 26-s − 27-s − 4·29-s + 4·30-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 + 1.78·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.26·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 1.03·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.894·20-s + 0.852·22-s − 0.834·23-s − 0.612·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s − 0.742·29-s + 0.730·30-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: $1$
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: \(0\)
Selberg data: \((2,\ 32487,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7814275153\)
\(L(\frac12)\) \(\approx\) \(0.7814275153\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.92803430802243, −14.41556384660158, −13.94598344407437, −13.27901375138402, −12.97895469555851, −12.76341928959910, −11.75974918619360, −11.00139646078963, −10.65478247852165, −10.04580971769586, −9.724527331279095, −9.374308559823552, −8.587819529830844, −8.130265949084457, −7.353342401113191, −6.867346849919181, −6.049046819337342, −5.516472003061428, −5.204791788076253, −4.561643356929584, −3.700837518835119, −2.659904825900388, −1.979125611878906, −1.480883016501761, −0.3888705876474981, 0.3888705876474981, 1.480883016501761, 1.979125611878906, 2.659904825900388, 3.700837518835119, 4.561643356929584, 5.204791788076253, 5.516472003061428, 6.049046819337342, 6.867346849919181, 7.353342401113191, 8.130265949084457, 8.587819529830844, 9.374308559823552, 9.724527331279095, 10.04580971769586, 10.65478247852165, 11.00139646078963, 11.75974918619360, 12.76341928959910, 12.97895469555851, 13.27901375138402, 13.94598344407437, 14.41556384660158, 14.92803430802243

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