Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7 \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  + 3-s + 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s − 2·17-s + 4·19-s − 21-s − 25-s + 27-s − 2·29-s + 8·31-s − 2·35-s − 6·37-s + 6·39-s − 10·41-s − 4·43-s + 2·45-s − 8·47-s + 49-s − 2·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s + 0.960·39-s − 1.56·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(162624\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{162624} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 162624,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.58986133450993, −13.13133955995909, −12.82249704107228, −12.01488016097310, −11.62631764915462, −11.14078608610893, −10.45498765835337, −10.15454890730323, −9.646785798512175, −9.216675178336634, −8.699338713961317, −8.304821697827577, −7.829143268829092, −7.098140172295971, −6.550419285285313, −6.240240144163529, −5.736877986963269, −5.041591296652738, −4.614166596635418, −3.741452321766847, −3.369198067647187, −2.940957130596175, −2.074022019461224, −1.606705114336533, −1.076040946316660, 0, 1.076040946316660, 1.606705114336533, 2.074022019461224, 2.940957130596175, 3.369198067647187, 3.741452321766847, 4.614166596635418, 5.041591296652738, 5.736877986963269, 6.240240144163529, 6.550419285285313, 7.098140172295971, 7.829143268829092, 8.304821697827577, 8.699338713961317, 9.216675178336634, 9.646785798512175, 10.15454890730323, 10.45498765835337, 11.14078608610893, 11.62631764915462, 12.01488016097310, 12.82249704107228, 13.13133955995909, 13.58986133450993

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