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 − 6·13-s + 2·17-s + 8·19-s + 8·23-s + 25-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s − 8·47-s + 10·53-s + 4·59-s − 2·61-s + 6·65-s + 4·67-s − 12·71-s + 2·73-s − 8·79-s − 4·83-s − 2·85-s − 6·89-s − 8·95-s + 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.520·59-s − 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 0.635·89-s − 0.820·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-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  =  $(2,\ 141120,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.347225134$
$L(\frac12)$  $\approx$  $2.347225134$
$L(\frac{3}{2})$   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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.27304418040579, −12.95720278476649, −12.40866736848789, −11.81282884760039, −11.65152333587092, −11.18579542347637, −10.38377155965304, −9.949388408853945, −9.710144453552129, −8.969461932616944, −8.680358513366737, −7.794257862742752, −7.553502638246126, −7.101626996186816, −6.708607775588003, −5.782062858016991, −5.378321409363272, −4.802821531416204, −4.542105556206608, −3.579417507562331, −3.118383112623330, −2.717138935055495, −1.909997468105602, −1.060092750642424, −0.5257566409606752, 0.5257566409606752, 1.060092750642424, 1.909997468105602, 2.717138935055495, 3.118383112623330, 3.579417507562331, 4.542105556206608, 4.802821531416204, 5.378321409363272, 5.782062858016991, 6.708607775588003, 7.101626996186816, 7.553502638246126, 7.794257862742752, 8.680358513366737, 8.969461932616944, 9.710144453552129, 9.949388408853945, 10.38377155965304, 11.18579542347637, 11.65152333587092, 11.81282884760039, 12.40866736848789, 12.95720278476649, 13.27304418040579

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