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

−17.21837655289706, −16.77159483165008, −15.88303029986168, −15.46868214947059, −14.79434021551051, −14.22947809000010, −14.03330839042900, −13.17312048188211, −12.77312551807090, −12.18869686120772, −11.50437905808672, −11.14554284370740, −10.06688753785745, −9.674579016854467, −8.907414450934320, −8.001996334534887, −7.400808285478021, −6.524632507009368, −5.995614380178735, −5.312495293385592, −4.688717700037229, −4.129593843906292, −3.225616757617063, −2.430153917800974, −1.734853020213135, 0, 1.734853020213135, 2.430153917800974, 3.225616757617063, 4.129593843906292, 4.688717700037229, 5.312495293385592, 5.995614380178735, 6.524632507009368, 7.400808285478021, 8.001996334534887, 8.907414450934320, 9.674579016854467, 10.06688753785745, 11.14554284370740, 11.50437905808672, 12.18869686120772, 12.77312551807090, 13.17312048188211, 14.03330839042900, 14.22947809000010, 14.79434021551051, 15.46868214947059, 15.88303029986168, 16.77159483165008, 17.21837655289706

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