Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 4·13-s + 16-s + 6·17-s − 4·19-s − 6·23-s − 5·25-s + 4·26-s + 6·29-s − 8·31-s − 32-s − 6·34-s − 10·37-s + 4·38-s − 6·41-s − 8·43-s + 6·46-s − 6·47-s + 5·50-s − 4·52-s − 6·58-s + 8·61-s + 8·62-s + 64-s − 4·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s + 0.784·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s + 0.884·46-s − 0.875·47-s + 0.707·50-s − 0.554·52-s − 0.787·58-s + 1.02·61-s + 1.01·62-s + 1/8·64-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(106722\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{106722} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 106722,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
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

−14.35232916706859, −13.69471386308613, −13.21651735936986, −12.45527940996748, −12.12404935343590, −11.88553182151534, −11.24568087627675, −10.47455624011694, −10.24194764763729, −9.784951126606731, −9.423728168672662, −8.547711086224653, −8.309902762996197, −7.797058215925956, −7.214604453259703, −6.791296070053912, −6.201057444577284, −5.487727711801215, −5.209094606600183, −4.392692740649945, −3.702829000635981, −3.232603061797970, −2.444328916939911, −1.834275457766010, −1.360277424921043, 0, 0, 1.360277424921043, 1.834275457766010, 2.444328916939911, 3.232603061797970, 3.702829000635981, 4.392692740649945, 5.209094606600183, 5.487727711801215, 6.201057444577284, 6.791296070053912, 7.214604453259703, 7.797058215925956, 8.309902762996197, 8.547711086224653, 9.423728168672662, 9.784951126606731, 10.24194764763729, 10.47455624011694, 11.24568087627675, 11.88553182151534, 12.12404935343590, 12.45527940996748, 13.21651735936986, 13.69471386308613, 14.35232916706859

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