Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s + 13-s + 14-s + 15-s + 16-s − 2·17-s − 18-s − 19-s − 20-s + 21-s − 4·23-s + 24-s + 25-s − 26-s − 27-s − 28-s + 2·29-s − 30-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 − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(51870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{51870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 51870,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
19 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.23129923017624, −14.57482981782321, −14.00386222593217, −13.21180358568269, −13.02049145077217, −12.23775504287759, −11.66368348992839, −11.60467355652453, −10.78624922973190, −10.25135107855380, −10.04569594646279, −9.228659330137271, −8.762900143688256, −8.185717990906224, −7.734787344998618, −6.985524876026676, −6.593779357501614, −6.111598626859794, −5.444184179753545, −4.700827862788747, −4.185259831336759, −3.354456271099381, −2.877961489805849, −1.811845723186736, −1.355199662134473, 0, 0, 1.355199662134473, 1.811845723186736, 2.877961489805849, 3.354456271099381, 4.185259831336759, 4.700827862788747, 5.444184179753545, 6.111598626859794, 6.593779357501614, 6.985524876026676, 7.734787344998618, 8.185717990906224, 8.762900143688256, 9.228659330137271, 10.04569594646279, 10.25135107855380, 10.78624922973190, 11.60467355652453, 11.66368348992839, 12.23775504287759, 13.02049145077217, 13.21180358568269, 14.00386222593217, 14.57482981782321, 15.23129923017624

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