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 − 4·7-s − 8-s − 10-s + 11-s + 2·13-s + 4·14-s + 16-s − 17-s + 2·19-s + 20-s − 22-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 − 2·38-s − 40-s + 12·41-s − 4·43-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 + 0.458·19-s + 0.223·20-s − 0.213·22-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 − 0.324·38-s − 0.158·40-s + 1.87·41-s − 0.609·43-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−16.30425889644692, −15.81477266255697, −15.22416991339147, −14.56913766671056, −13.93756008818878, −13.28130784596763, −12.85896415343838, −12.38173620629775, −11.61337863978241, −10.93466901077246, −10.59394591415813, −9.690588160074475, −9.358283224732938, −9.162251996897190, −8.178414337323268, −7.594763818956332, −6.842334937200874, −6.389923787052793, −5.879215667273590, −5.188852156754421, −4.040601845593685, −3.467027151059009, −2.769883639157979, −1.951080960224562, −1.000699285954614, 0, 1.000699285954614, 1.951080960224562, 2.769883639157979, 3.467027151059009, 4.040601845593685, 5.188852156754421, 5.879215667273590, 6.389923787052793, 6.842334937200874, 7.594763818956332, 8.178414337323268, 9.162251996897190, 9.358283224732938, 9.690588160074475, 10.59394591415813, 10.93466901077246, 11.61337863978241, 12.38173620629775, 12.85896415343838, 13.28130784596763, 13.93756008818878, 14.56913766671056, 15.22416991339147, 15.81477266255697, 16.30425889644692

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