Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \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  + 2·5-s + 7-s − 6·13-s + 2·17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 2·35-s + 6·37-s + 10·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s + 4·59-s + 10·61-s − 12·65-s + 12·67-s − 2·73-s + 16·79-s − 4·83-s + 4·85-s − 18·89-s − 6·91-s + 8·95-s + 2·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 1.48·65-s + 1.46·67-s − 0.234·73-s + 1.80·79-s − 0.439·83-s + 0.433·85-s − 1.90·89-s − 0.628·91-s + 0.820·95-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(121968\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{121968} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 121968,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
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

−14.00678924981032, −13.12088689214424, −12.91488807053747, −12.47095115671613, −11.76191706723076, −11.45041711046954, −10.93139911777242, −10.24648055377529, −9.779995617140661, −9.496547744366536, −9.180650644440211, −8.265287557098288, −7.822425835260483, −7.400051221365242, −6.895949817327239, −6.248773303587946, −5.614322815659625, −5.245704649337687, −4.876248692313540, −4.076787086606947, −3.495577389626506, −2.702187733382244, −2.268777308618139, −1.682598187477938, −0.9342217557058439, 0, 0.9342217557058439, 1.682598187477938, 2.268777308618139, 2.702187733382244, 3.495577389626506, 4.076787086606947, 4.876248692313540, 5.245704649337687, 5.614322815659625, 6.248773303587946, 6.895949817327239, 7.400051221365242, 7.822425835260483, 8.265287557098288, 9.180650644440211, 9.496547744366536, 9.779995617140661, 10.24648055377529, 10.93139911777242, 11.45041711046954, 11.76191706723076, 12.47095115671613, 12.91488807053747, 13.12088689214424, 14.00678924981032

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