Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{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  − 5-s + 2·13-s − 6·17-s − 8·19-s + 25-s + 6·29-s − 4·31-s + 10·37-s − 6·41-s − 4·43-s − 6·53-s − 12·59-s − 10·61-s − 2·65-s − 4·67-s + 12·71-s + 10·73-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s + 8·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.554·13-s − 1.45·17-s − 1.83·19-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{141120} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 141120,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6239411912\)
\(L(\frac12)\)  \(\approx\)  \(0.6239411912\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.50665509308306, −12.84317999836061, −12.54158507637124, −12.00888290986052, −11.32840104277169, −11.05053932824814, −10.61976809459832, −10.19959549996960, −9.334858861011366, −9.067647042663900, −8.533179351807103, −8.018897896798755, −7.727424483753925, −6.764947429923419, −6.491458290069189, −6.234911197112835, −5.358472544315775, −4.671454282541451, −4.374165953967120, −3.850262516820304, −3.149508935800491, −2.513000941576583, −1.938682038139110, −1.240261894334252, −0.2369603868099414, 0.2369603868099414, 1.240261894334252, 1.938682038139110, 2.513000941576583, 3.149508935800491, 3.850262516820304, 4.374165953967120, 4.671454282541451, 5.358472544315775, 6.234911197112835, 6.491458290069189, 6.764947429923419, 7.727424483753925, 8.018897896798755, 8.533179351807103, 9.067647042663900, 9.334858861011366, 10.19959549996960, 10.61976809459832, 11.05053932824814, 11.32840104277169, 12.00888290986052, 12.54158507637124, 12.84317999836061, 13.50665509308306

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