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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 11-s + 2·13-s + 4·14-s + 16-s + 17-s − 4·19-s + 20-s − 22-s − 6·23-s + 25-s − 2·26-s − 4·28-s + 6·29-s − 4·31-s − 32-s − 34-s − 4·35-s + 2·37-s + 4·38-s − 40-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.676·35-s + 0.328·37-s + 0.648·38-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  =  0
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.012443698$
$L(\frac12)$  $\approx$  $1.012443698$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.17425634608489, −15.55497410316430, −14.86795080978177, −14.23318888648749, −13.65995882588987, −13.06746741522754, −12.51028789335184, −12.11306006943501, −11.32456692172967, −10.58346563194761, −10.22046279818612, −9.717225504585124, −8.984611337664621, −8.803117829558560, −7.860504902733317, −7.264413805804673, −6.527511238920394, −6.039931847135582, −5.784801665636106, −4.466396035738850, −3.852421971763558, −3.031699357314842, −2.421532200908066, −1.492224588367505, −0.4832898586073334, 0.4832898586073334, 1.492224588367505, 2.421532200908066, 3.031699357314842, 3.852421971763558, 4.466396035738850, 5.784801665636106, 6.039931847135582, 6.527511238920394, 7.264413805804673, 7.860504902733317, 8.803117829558560, 8.984611337664621, 9.717225504585124, 10.22046279818612, 10.58346563194761, 11.32456692172967, 12.11306006943501, 12.51028789335184, 13.06746741522754, 13.65995882588987, 14.23318888648749, 14.86795080978177, 15.55497410316430, 16.17425634608489

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