Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 47^{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 + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s + 20-s − 4·21-s − 24-s + 25-s + 2·26-s + 27-s − 4·28-s + 6·29-s − 30-s − 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.182·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(66270\)    =    \(2 \cdot 3 \cdot 5 \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{66270} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 66270,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.027626265\)
\(L(\frac12)\)  \(\approx\)  \(2.027626265\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;47\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
47 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.36972948845267, −13.75892148206178, −13.10936822480755, −12.67121641694010, −12.27994800327712, −11.80210123087959, −10.98021010669754, −10.37519615658859, −10.02904637626694, −9.530917344164382, −9.244862099975577, −8.799187583731888, −7.857436773324824, −7.567299722238628, −7.133609447317155, −6.262826467329202, −6.108778168436033, −5.310744532376230, −4.686580398321536, −3.552720831357550, −3.390913465949447, −2.710814426077753, −2.139692820973244, −1.198964284846529, −0.5605926808706801, 0.5605926808706801, 1.198964284846529, 2.139692820973244, 2.710814426077753, 3.390913465949447, 3.552720831357550, 4.686580398321536, 5.310744532376230, 6.108778168436033, 6.262826467329202, 7.133609447317155, 7.567299722238628, 7.857436773324824, 8.799187583731888, 9.244862099975577, 9.530917344164382, 10.02904637626694, 10.37519615658859, 10.98021010669754, 11.80210123087959, 12.27994800327712, 12.67121641694010, 13.10936822480755, 13.75892148206178, 14.36972948845267

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