Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 \cdot 11^{2} $
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 − 2·3-s + 4-s − 5-s + 2·6-s − 7-s − 8-s + 9-s + 10-s − 2·12-s + 4·13-s + 14-s + 2·15-s + 16-s − 18-s + 4·19-s − 20-s + 2·21-s + 2·24-s + 25-s − 4·26-s + 4·27-s − 28-s + 6·29-s − 2·30-s − 10·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.436·21-s + 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 0.365·30-s − 1.79·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8470,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7450435455$
$L(\frac12)$  $\approx$  $0.7450435455$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.94208542117468, −16.24925206914574, −15.94752491222534, −15.63320298169952, −14.57156070317032, −14.09762895172135, −13.20375886701535, −12.52670008931359, −12.10958055902515, −11.38180681254203, −10.93124610183332, −10.63528438285353, −9.687282318648578, −9.140188855829312, −8.482736486289717, −7.694813613199627, −7.139014066857804, −6.342231993353363, −5.889343868482678, −5.239996646484318, −4.272137175003740, −3.472964685537278, −2.613054780788344, −1.284593679486001, −0.5667571043868801, 0.5667571043868801, 1.284593679486001, 2.613054780788344, 3.472964685537278, 4.272137175003740, 5.239996646484318, 5.889343868482678, 6.342231993353363, 7.139014066857804, 7.694813613199627, 8.482736486289717, 9.140188855829312, 9.687282318648578, 10.63528438285353, 10.93124610183332, 11.38180681254203, 12.10958055902515, 12.52670008931359, 13.20375886701535, 14.09762895172135, 14.57156070317032, 15.63320298169952, 15.94752491222534, 16.24925206914574, 16.94208542117468

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