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 2

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 − 2·19-s − 20-s + 22-s − 6·23-s + 25-s + 4·26-s − 2·28-s + 2·29-s − 8·31-s − 32-s + 34-s + 2·35-s − 6·37-s + 2·38-s + 40-s − 2·41-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 − 0.458·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.986·37-s + 0.324·38-s + 0.158·40-s − 0.312·41-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  =  2
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 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−16.42106038859855, −16.02338065197462, −15.44433289105682, −14.93227196496853, −14.38509111313134, −13.67647646709317, −12.99731815494733, −12.40918530621821, −12.06600004938545, −11.40338055163840, −10.65149213947049, −10.29571405387176, −9.577467535042645, −9.248564632081936, −8.392522359979138, −7.935667961311850, −7.322388869450150, −6.693660145079734, −6.215504047415622, −5.305710318950527, −4.688637859250361, −3.748299468021743, −3.169260855446143, −2.315247360662828, −1.609294790666292, 0, 0, 1.609294790666292, 2.315247360662828, 3.169260855446143, 3.748299468021743, 4.688637859250361, 5.305710318950527, 6.215504047415622, 6.693660145079734, 7.322388869450150, 7.935667961311850, 8.392522359979138, 9.248564632081936, 9.577467535042645, 10.29571405387176, 10.65149213947049, 11.40338055163840, 12.06600004938545, 12.40918530621821, 12.99731815494733, 13.67647646709317, 14.38509111313134, 14.93227196496853, 15.44433289105682, 16.02338065197462, 16.42106038859855

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