Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{2} $
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 − 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 + 12·59-s + 2·61-s − 4·65-s − 10·67-s − 12·71-s + 16·73-s + 8·79-s − 6·89-s + 16·91-s + 4·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-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 + 1.56·59-s + 0.256·61-s − 0.496·65-s − 1.22·67-s − 1.42·71-s + 1.87·73-s + 0.900·79-s − 0.635·89-s + 1.67·91-s + 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(348480\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{348480} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 348480,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
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

−12.75924385488624, −12.36174833082711, −11.89933185036684, −11.67130208312190, −10.72863925429913, −10.47720433650878, −9.916033132015611, −9.695396416446947, −9.224504695830435, −8.905015805176085, −8.056721338242858, −7.780332857179650, −6.995066492459269, −6.896309236629787, −6.290688295512167, −5.722472426990258, −5.451452234037214, −4.795271277937662, −4.228050655642102, −3.558694331137954, −3.242929776261448, −2.555202294375403, −2.211864904742156, −1.435740023285180, −0.6098068506314400, 0, 0.6098068506314400, 1.435740023285180, 2.211864904742156, 2.555202294375403, 3.242929776261448, 3.558694331137954, 4.228050655642102, 4.795271277937662, 5.451452234037214, 5.722472426990258, 6.290688295512167, 6.896309236629787, 6.995066492459269, 7.780332857179650, 8.056721338242858, 8.905015805176085, 9.224504695830435, 9.695396416446947, 9.916033132015611, 10.47720433650878, 10.72863925429913, 11.67130208312190, 11.89933185036684, 12.36174833082711, 12.75924385488624

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