Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s + 11-s + 6·13-s + 14-s − 16-s − 2·17-s + 4·19-s − 2·20-s + 22-s − 25-s + 6·26-s − 28-s + 2·29-s + 8·31-s + 5·32-s − 2·34-s + 2·35-s + 6·37-s + 4·38-s − 6·40-s − 10·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s − 0.948·40-s − 1.56·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{693} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 693,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.282665969$
$L(\frac12)$  $\approx$  $2.282665969$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
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

−19.69831175842644, −18.52389602822777, −18.20924659706994, −17.53805537566189, −16.78336698131699, −15.68032163506511, −15.15071898747602, −13.99184551628743, −13.73993735305562, −13.24007776433405, −12.15218092897343, −11.45629807941597, −10.43992262323558, −9.532486316872858, −8.828553587531721, −8.022234507412908, −6.484701212309219, −5.943821735183565, −4.992342911753722, −4.043786122880624, −2.957670798231812, −1.349554964020272, 1.349554964020272, 2.957670798231812, 4.043786122880624, 4.992342911753722, 5.943821735183565, 6.484701212309219, 8.022234507412908, 8.828553587531721, 9.532486316872858, 10.43992262323558, 11.45629807941597, 12.15218092897343, 13.24007776433405, 13.73993735305562, 13.99184551628743, 15.15071898747602, 15.68032163506511, 16.78336698131699, 17.53805537566189, 18.20924659706994, 18.52389602822777, 19.69831175842644

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