Properties

Degree $2$
Conductor $86190$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 5·11-s − 12-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s − 5·22-s − 24-s + 25-s − 27-s + 28-s − 2·29-s + 30-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.218·21-s − 1.06·22-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + 0.176·32-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.09565553576492, −13.56544661445026, −13.04661834043446, −12.84507091725308, −12.08087624638954, −11.79976920101842, −11.27323694541830, −10.70945826202389, −10.48460742263904, −9.803349541631352, −9.258709003821877, −8.359683530225462, −8.029893124822188, −7.452090047202514, −7.130434224506674, −6.295169487962764, −5.871071332100893, −5.234822774643279, −4.910935782484990, −4.357527385303614, −3.686347132250452, −3.038181740660724, −2.461536765413145, −1.728794897286561, −0.8594998122406860, 0, 0.8594998122406860, 1.728794897286561, 2.461536765413145, 3.038181740660724, 3.686347132250452, 4.357527385303614, 4.910935782484990, 5.234822774643279, 5.871071332100893, 6.295169487962764, 7.130434224506674, 7.452090047202514, 8.029893124822188, 8.359683530225462, 9.258709003821877, 9.803349541631352, 10.48460742263904, 10.70945826202389, 11.27323694541830, 11.79976920101842, 12.08087624638954, 12.84507091725308, 13.04661834043446, 13.56544661445026, 14.09565553576492

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