Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{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  − 3-s − 5-s + 7-s + 9-s + 15-s + 6·17-s − 4·19-s − 21-s + 25-s − 27-s + 6·29-s − 4·31-s − 35-s + 10·37-s − 6·41-s − 8·43-s − 45-s + 49-s − 6·51-s − 6·53-s + 4·57-s − 12·59-s + 14·61-s + 63-s − 4·67-s − 2·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(283920\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{283920} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 283920,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
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

−12.90772005208802, −12.42720620803060, −12.01299692027884, −11.59228554293219, −11.22426870272326, −10.70908999700293, −10.22328532295311, −9.891272458434288, −9.365963909161193, −8.660632224336367, −8.280615966269825, −7.863551899172894, −7.404519573411846, −6.801437381816584, −6.413990823529794, −5.819290064991061, −5.386207955382366, −4.841518752894644, −4.359240072518913, −3.926408861242010, −3.166271180293409, −2.833077371706706, −1.892626107984032, −1.406801732114246, −0.7381882169339960, 0, 0.7381882169339960, 1.406801732114246, 1.892626107984032, 2.833077371706706, 3.166271180293409, 3.926408861242010, 4.359240072518913, 4.841518752894644, 5.386207955382366, 5.819290064991061, 6.413990823529794, 6.801437381816584, 7.404519573411846, 7.863551899172894, 8.280615966269825, 8.660632224336367, 9.365963909161193, 9.891272458434288, 10.22328532295311, 10.70908999700293, 11.22426870272326, 11.59228554293219, 12.01299692027884, 12.42720620803060, 12.90772005208802

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