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 − 2·7-s − 8-s + 10-s + 11-s − 4·13-s + 2·14-s + 16-s − 17-s − 6·19-s − 20-s − 22-s + 2·23-s + 25-s + 4·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s + 34-s + 2·35-s − 2·37-s + 6·38-s + 40-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.223·20-s − 0.213·22-s + 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.328·37-s + 0.973·38-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.14940963265893, −15.73587694692979, −15.18571075730023, −14.61036505661765, −14.14939150200877, −13.27700145455612, −12.59913386197760, −12.36922151787253, −11.72922067864808, −10.93857836227078, −10.57174936327552, −9.963947232818078, −9.262721541814381, −8.907620356398440, −8.258537399151443, −7.429222063743404, −7.146386263185677, −6.365071034562125, −5.916800098580665, −4.863771799678897, −4.285633841193968, −3.474730280237911, −2.662543620740451, −2.089195921859799, −0.8554177841824049, 0, 0.8554177841824049, 2.089195921859799, 2.662543620740451, 3.474730280237911, 4.285633841193968, 4.863771799678897, 5.916800098580665, 6.365071034562125, 7.146386263185677, 7.429222063743404, 8.258537399151443, 8.907620356398440, 9.262721541814381, 9.963947232818078, 10.57174936327552, 10.93857836227078, 11.72922067864808, 12.36922151787253, 12.59913386197760, 13.27700145455612, 14.14939150200877, 14.61036505661765, 15.18571075730023, 15.73587694692979, 16.14940963265893

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