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  − 2·2-s + 2·4-s − 5-s + 7-s + 2·10-s − 4·13-s − 2·14-s − 4·16-s + 17-s − 2·20-s + 4·23-s − 4·25-s + 8·26-s + 2·28-s − 2·31-s + 8·32-s − 2·34-s − 35-s + 6·37-s − 2·41-s − 3·43-s − 8·46-s + 7·47-s + 49-s + 8·50-s − 8·52-s + 12·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s + 0.632·10-s − 1.10·13-s − 0.534·14-s − 16-s + 0.242·17-s − 0.447·20-s + 0.834·23-s − 4/5·25-s + 1.56·26-s + 0.377·28-s − 0.359·31-s + 1.41·32-s − 0.342·34-s − 0.169·35-s + 0.986·37-s − 0.312·41-s − 0.457·43-s − 1.17·46-s + 1.02·47-s + 1/7·49-s + 1.13·50-s − 1.10·52-s + 1.64·53-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 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−17.29263131782650, −16.95812599475986, −16.40945860605437, −15.77308060080291, −15.03044550768248, −14.74629901376987, −13.76854044040676, −13.30326867480311, −12.32227295363948, −11.88384946167525, −11.22023228837428, −10.63143162609619, −10.06503457726896, −9.401614664894218, −8.941092468535752, −8.186653815651648, −7.582065985049912, −7.294582507355323, −6.435514626539825, −5.431571406725859, −4.696462983861773, −3.953942977847086, −2.808521166198442, −2.012899913917804, −1.018219195911087, 0, 1.018219195911087, 2.012899913917804, 2.808521166198442, 3.953942977847086, 4.696462983861773, 5.431571406725859, 6.435514626539825, 7.294582507355323, 7.582065985049912, 8.186653815651648, 8.941092468535752, 9.401614664894218, 10.06503457726896, 10.63143162609619, 11.22023228837428, 11.88384946167525, 12.32227295363948, 13.30326867480311, 13.76854044040676, 14.74629901376987, 15.03044550768248, 15.77308060080291, 16.40945860605437, 16.95812599475986, 17.29263131782650

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