Properties

Degree $2$
Conductor $86190$
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 − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 2·14-s + 15-s + 16-s + 17-s − 18-s − 20-s − 2·21-s − 4·22-s + 8·23-s + 24-s + 25-s − 27-s + 2·28-s − 8·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{86190} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 86190,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.36745839551061, −13.63541207212651, −13.07728493845405, −12.51684489136846, −11.98849643459839, −11.55333390179349, −11.16326197669960, −10.93308618305798, −10.09443094821537, −9.745356402337775, −9.003809049591944, −8.783326271835773, −8.080674183277893, −7.588976115879501, −7.065437691966549, −6.626513617966400, −6.048949207046494, −5.411222067622575, −4.760328651533251, −4.357647336659294, −3.540048135739730, −3.049963425297531, −2.081960321094442, −1.364253295131277, −0.9820015724674870, 0, 0.9820015724674870, 1.364253295131277, 2.081960321094442, 3.049963425297531, 3.540048135739730, 4.357647336659294, 4.760328651533251, 5.411222067622575, 6.048949207046494, 6.626513617966400, 7.065437691966549, 7.588976115879501, 8.080674183277893, 8.783326271835773, 9.003809049591944, 9.745356402337775, 10.09443094821537, 10.93308618305798, 11.16326197669960, 11.55333390179349, 11.98849643459839, 12.51684489136846, 13.07728493845405, 13.63541207212651, 14.36745839551061

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