Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
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 + 3·5-s + 7-s + 3·8-s − 3·10-s + 7·13-s − 14-s − 16-s − 3·17-s − 2·19-s − 3·20-s + 4·23-s + 4·25-s − 7·26-s − 28-s − 7·29-s − 10·31-s − 5·32-s + 3·34-s + 3·35-s + 37-s + 2·38-s + 9·40-s + 5·41-s + 6·43-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.377·7-s + 1.06·8-s − 0.948·10-s + 1.94·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.670·20-s + 0.834·23-s + 4/5·25-s − 1.37·26-s − 0.188·28-s − 1.29·29-s − 1.79·31-s − 0.883·32-s + 0.514·34-s + 0.507·35-s + 0.164·37-s + 0.324·38-s + 1.42·40-s + 0.780·41-s + 0.914·43-s − 0.589·46-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.840650925$
$L(\frac12)$  $\approx$  $1.840650925$
$L(\frac{3}{2})$   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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−17.22157332029561, −16.58635286261194, −16.20039385190507, −15.26974183393369, −14.64639084284254, −13.98463092795742, −13.46955683547422, −13.11197981516680, −12.61552356214521, −11.27912190300213, −10.86705942206540, −10.58334857349274, −9.531526305746704, −9.194482563157440, −8.754680770725522, −8.092694130084279, −7.236240802945193, −6.519558914634571, −5.623517317005090, −5.391632921970673, −4.201053709414410, −3.702135811506423, −2.301258163815990, −1.653740192594963, −0.8137366007658102, 0.8137366007658102, 1.653740192594963, 2.301258163815990, 3.702135811506423, 4.201053709414410, 5.391632921970673, 5.623517317005090, 6.519558914634571, 7.236240802945193, 8.092694130084279, 8.754680770725522, 9.194482563157440, 9.531526305746704, 10.58334857349274, 10.86705942206540, 11.27912190300213, 12.61552356214521, 13.11197981516680, 13.46955683547422, 13.98463092795742, 14.64639084284254, 15.26974183393369, 16.20039385190507, 16.58635286261194, 17.22157332029561

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