Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 29^{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-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s + 11-s − 12-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 18-s + 4·19-s − 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s − 4·26-s − 27-s + 2·28-s − 8·31-s + 32-s − 33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55506\)    =    \(2 \cdot 3 \cdot 11 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{55506} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 55506,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
29 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.76072840984293, −14.22212588624248, −13.64518272326291, −13.10142758249823, −12.57044448212356, −12.00093017509604, −11.75249288077087, −11.24254834379269, −10.73638463880211, −10.05024274379191, −9.619487767986583, −9.142157184497605, −8.108697038770464, −7.795152873884944, −7.168349024209767, −6.863686966384258, −5.878914158924880, −5.525790670424890, −5.079368752526287, −4.548663303186098, −3.865103376818500, −3.187026409854268, −2.585339559261515, −1.583765024752358, −1.208608238018424, 0, 1.208608238018424, 1.583765024752358, 2.585339559261515, 3.187026409854268, 3.865103376818500, 4.548663303186098, 5.079368752526287, 5.525790670424890, 5.878914158924880, 6.863686966384258, 7.168349024209767, 7.795152873884944, 8.108697038770464, 9.142157184497605, 9.619487767986583, 10.05024274379191, 10.73638463880211, 11.24254834379269, 11.75249288077087, 12.00093017509604, 12.57044448212356, 13.10142758249823, 13.64518272326291, 14.22212588624248, 14.76072840984293

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