Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·7-s − 11-s − 4·13-s + 4·19-s − 6·23-s + 25-s + 6·29-s − 8·31-s − 4·35-s + 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s + 6·53-s + 55-s − 12·59-s + 2·61-s + 4·65-s + 10·67-s − 12·71-s − 16·73-s − 4·77-s − 8·79-s − 6·89-s − 16·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.10·13-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.496·65-s + 1.22·67-s − 1.42·71-s − 1.87·73-s − 0.455·77-s − 0.900·79-s − 0.635·89-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7920} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 7920,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−17.37354631583714, −16.75027828602611, −16.05728731637047, −15.56033313829902, −14.73033179117001, −14.57152867223702, −13.87813601018116, −13.27442689761422, −12.32023615405222, −11.90202502397731, −11.53436965301769, −10.71988682564034, −10.19417978371576, −9.513137876021492, −8.592827101400171, −8.183040109344344, −7.429580018146863, −7.178857054143156, −5.977661504503612, −5.240430836716556, −4.743993085060654, −4.061295466192231, −3.067718089789004, −2.159714183077436, −1.354228979326143, 0, 1.354228979326143, 2.159714183077436, 3.067718089789004, 4.061295466192231, 4.743993085060654, 5.240430836716556, 5.977661504503612, 7.178857054143156, 7.429580018146863, 8.183040109344344, 8.592827101400171, 9.513137876021492, 10.19417978371576, 10.71988682564034, 11.53436965301769, 11.90202502397731, 12.32023615405222, 13.27442689761422, 13.87813601018116, 14.57152867223702, 14.73033179117001, 15.56033313829902, 16.05728731637047, 16.75027828602611, 17.37354631583714

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