Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17^{2} $
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 + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s − 7·19-s + 20-s − 22-s − 3·23-s + 25-s − 26-s − 28-s − 8·29-s + 5·31-s − 32-s − 35-s − 5·37-s + 7·38-s − 40-s − 10·41-s + 2·43-s + 44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.60·19-s + 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.48·29-s + 0.898·31-s − 0.176·32-s − 0.169·35-s − 0.821·37-s + 1.13·38-s − 0.158·40-s − 1.56·41-s + 0.304·43-s + 0.150·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(286110\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{286110} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 286110,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.04214766590461, −12.43886716990282, −12.05738365554378, −11.38792495499901, −11.17801333401427, −10.46484757370320, −10.20919319758852, −9.797456452536018, −9.211660779749745, −8.838113730130296, −8.439517140026765, −7.892935247365018, −7.463143623698239, −6.706730014897277, −6.401493574137261, −6.185875761986189, −5.396006662880862, −4.957590288499851, −4.241179176196488, −3.659862073752530, −3.264000422971378, −2.453987822476451, −1.913522419018372, −1.605806425544222, −0.6418492529897764, 0, 0.6418492529897764, 1.605806425544222, 1.913522419018372, 2.453987822476451, 3.264000422971378, 3.659862073752530, 4.241179176196488, 4.957590288499851, 5.396006662880862, 6.185875761986189, 6.401493574137261, 6.706730014897277, 7.463143623698239, 7.892935247365018, 8.439517140026765, 8.838113730130296, 9.211660779749745, 9.797456452536018, 10.20919319758852, 10.46484757370320, 11.17801333401427, 11.38792495499901, 12.05738365554378, 12.43886716990282, 13.04214766590461

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