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 + 3·7-s − 8-s + 10-s + 11-s − 3·14-s + 16-s − 17-s − 8·19-s − 20-s − 22-s − 3·23-s + 25-s + 3·28-s − 3·29-s + 3·31-s − 32-s + 34-s − 3·35-s + 10·37-s + 8·38-s + 40-s − 43-s + 44-s + 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.566·28-s − 0.557·29-s + 0.538·31-s − 0.176·32-s + 0.171·34-s − 0.507·35-s + 1.64·37-s + 1.29·38-s + 0.158·40-s − 0.152·43-s + 0.150·44-s + 0.442·46-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−16.21386006726581, −15.66005832367697, −15.02268194978313, −14.64080458976736, −14.25447463362214, −13.27929096120459, −12.82832606077575, −12.09430816778778, −11.59409998109451, −10.94865996477720, −10.84724230255871, −9.862597933763781, −9.428306565451559, −8.547013180729491, −8.190157153596190, −7.875973486012057, −6.949157308157672, −6.469567022584722, −5.760452063679026, −4.858642658883683, −4.290491980501828, −3.668265419550367, −2.501726040536834, −1.973368487368723, −1.078652642192738, 0, 1.078652642192738, 1.973368487368723, 2.501726040536834, 3.668265419550367, 4.290491980501828, 4.858642658883683, 5.760452063679026, 6.469567022584722, 6.949157308157672, 7.875973486012057, 8.190157153596190, 8.547013180729491, 9.428306565451559, 9.862597933763781, 10.84724230255871, 10.94865996477720, 11.59409998109451, 12.09430816778778, 12.82832606077575, 13.27929096120459, 14.25447463362214, 14.64080458976736, 15.02268194978313, 15.66005832367697, 16.21386006726581

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