Properties

Degree 2
Conductor $ 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-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 6·13-s + 14-s − 16-s + 2·17-s − 4·19-s − 2·20-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 + 8·47-s + 49-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 − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.447·20-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 + 1.16·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 \)
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

−17.37599172853484, −17.07893313019875, −16.40395482956638, −15.74940126786307, −14.82390783889958, −14.44359534650089, −13.82964827861948, −13.23525683399496, −12.64734573450950, −12.17407212317209, −11.21471210447779, −10.44285725572252, −9.972711844308023, −9.544252347607646, −9.087313765474909, −8.234519179482913, −7.616737848259141, −7.028812956085635, −6.060609116667613, −5.560100739453676, −4.594425002133160, −4.174622622694998, −2.809050266628644, −2.213325538664651, −1.109238431754849, 0, 1.109238431754849, 2.213325538664651, 2.809050266628644, 4.174622622694998, 4.594425002133160, 5.560100739453676, 6.060609116667613, 7.028812956085635, 7.616737848259141, 8.234519179482913, 9.087313765474909, 9.544252347607646, 9.972711844308023, 10.44285725572252, 11.21471210447779, 12.17407212317209, 12.64734573450950, 13.23525683399496, 13.82964827861948, 14.44359534650089, 14.82390783889958, 15.74940126786307, 16.40395482956638, 17.07893313019875, 17.37599172853484

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