Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17 $
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 − 4·13-s + 14-s + 16-s + 17-s + 2·19-s − 20-s − 22-s + 3·23-s + 25-s + 4·26-s − 28-s − 3·29-s + 5·31-s − 32-s − 34-s + 35-s + 8·37-s − 2·38-s + 40-s + 6·41-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 − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.223·20-s − 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.171·34-s + 0.169·35-s + 1.31·37-s − 0.324·38-s + 0.158·40-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{16830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
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

−16.36430175831325, −15.60310498064062, −15.22446627380818, −14.57794695774520, −14.17153377882042, −13.27842549772963, −12.75385403542787, −12.18474031623375, −11.64353093688772, −11.18385408067396, −10.46373897838743, −9.856985383522023, −9.420407021838762, −8.894453875063968, −8.118126199414686, −7.469494152253441, −7.259680754906345, −6.301412304945833, −5.895129897139174, −4.814011969069923, −4.428919380926584, −3.244960946055418, −2.932291116254940, −1.905446954895626, −0.9600851438545617, 0, 0.9600851438545617, 1.905446954895626, 2.932291116254940, 3.244960946055418, 4.428919380926584, 4.814011969069923, 5.895129897139174, 6.301412304945833, 7.259680754906345, 7.469494152253441, 8.118126199414686, 8.894453875063968, 9.420407021838762, 9.856985383522023, 10.46373897838743, 11.18385408067396, 11.64353093688772, 12.18474031623375, 12.75385403542787, 13.27842549772963, 14.17153377882042, 14.57794695774520, 15.22446627380818, 15.60310498064062, 16.36430175831325

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