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  + 1.36·2-s − 0.129·4-s − 0.636·5-s − 7-s − 2.91·8-s − 0.870·10-s + 3.59·13-s − 1.36·14-s − 3.72·16-s + 1.46·17-s + 5.85·19-s + 0.0824·20-s − 8.56·23-s − 4.59·25-s + 4.91·26-s + 0.129·28-s + 7.73·29-s − 9.44·31-s + 0.731·32-s + 2·34-s + 0.636·35-s + 5.59·37-s + 8.00·38-s + 1.85·40-s + 8.37·41-s − 1.74·43-s − 11.7·46-s + ⋯
L(s)  = 1  + 0.967·2-s − 0.0647·4-s − 0.284·5-s − 0.377·7-s − 1.02·8-s − 0.275·10-s + 0.997·13-s − 0.365·14-s − 0.931·16-s + 0.354·17-s + 1.34·19-s + 0.0184·20-s − 1.78·23-s − 0.918·25-s + 0.964·26-s + 0.0244·28-s + 1.43·29-s − 1.69·31-s + 0.129·32-s + 0.342·34-s + 0.107·35-s + 0.919·37-s + 1.29·38-s + 0.293·40-s + 1.30·41-s − 0.265·43-s − 1.72·46-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  :  $\chi_{7623} (1, \cdot )$
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 \( 1 - 1.36T + 2T^{2} \)
5 \( 1 + 0.636T + 5T^{2} \)
13 \( 1 - 3.59T + 13T^{2} \)
17 \( 1 - 1.46T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 + 8.56T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 5.59T + 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 + 1.74T + 43T^{2} \)
47 \( 1 - 4.99T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 + 9.66T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 + 4.19T + 89T^{2} \)
97 \( 1 - 7.44T + 97T^{2} \)
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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

−7.58000823918217821749322394399, −6.58861878836160112417024385309, −5.79329499176585255468858951346, −5.66355187675993613903478403658, −4.50828283615525444723039872182, −3.95752538257071210501812986912, −3.37668536365208490872264852948, −2.59206584808179450413215240631, −1.29361731405575613553715967331, 0, 1.29361731405575613553715967331, 2.59206584808179450413215240631, 3.37668536365208490872264852948, 3.95752538257071210501812986912, 4.50828283615525444723039872182, 5.66355187675993613903478403658, 5.79329499176585255468858951346, 6.58861878836160112417024385309, 7.58000823918217821749322394399

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