Properties

Label 2-32487-1.1-c1-0-3
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  + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s + 9-s − 4·10-s + 5·11-s − 2·12-s − 13-s + 2·15-s − 4·16-s − 17-s + 2·18-s − 6·19-s − 4·20-s + 10·22-s − 4·23-s − 25-s − 2·26-s − 27-s + 8·29-s + 4·30-s + 31-s − 8·32-s − 5·33-s − 2·34-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 + 1.50·11-s − 0.577·12-s − 0.277·13-s + 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.37·19-s − 0.894·20-s + 2.13·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.48·29-s + 0.730·30-s + 0.179·31-s − 1.41·32-s − 0.870·33-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: \(0\)
Selberg data: \((2,\ 32487,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.276620091\)
\(L(\frac12)\) \(\approx\) \(2.276620091\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.79789729039036, −14.65395756142947, −13.93576496748881, −13.55499756154794, −12.78978939930530, −12.30769732772658, −11.96664504762622, −11.69039634230206, −10.99136856386567, −10.54168071438841, −9.727445973125200, −9.065329753915546, −8.521912573548510, −7.836721245978190, −7.049201562384342, −6.589551469748986, −6.112079676551136, −5.635582846337481, −4.605589963104482, −4.336704377012972, −4.018684266431482, −3.247317924433207, −2.455301975108319, −1.588588304597067, −0.4627658944599375, 0.4627658944599375, 1.588588304597067, 2.455301975108319, 3.247317924433207, 4.018684266431482, 4.336704377012972, 4.605589963104482, 5.635582846337481, 6.112079676551136, 6.589551469748986, 7.049201562384342, 7.836721245978190, 8.521912573548510, 9.065329753915546, 9.727445973125200, 10.54168071438841, 10.99136856386567, 11.69039634230206, 11.96664504762622, 12.30769732772658, 12.78978939930530, 13.55499756154794, 13.93576496748881, 14.65395756142947, 14.79789729039036

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