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  − 2·4-s + 3·5-s + 7-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s + 4·25-s − 2·28-s + 6·29-s + 2·31-s + 3·35-s − 10·37-s − 6·41-s + 11·43-s + 3·47-s + 49-s + 8·52-s − 12·53-s + 3·59-s + 8·61-s − 8·64-s − 12·65-s + 5·67-s − 6·68-s + 12·71-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 0.377·7-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s + 4/5·25-s − 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 1.64·37-s − 0.937·41-s + 1.67·43-s + 0.437·47-s + 1/7·49-s + 1.10·52-s − 1.64·53-s + 0.390·59-s + 1.02·61-s − 64-s − 1.48·65-s + 0.610·67-s − 0.727·68-s + 1.42·71-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.141285193$
$L(\frac12)$  $\approx$  $2.141285193$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−17.34862255893244, −16.80861786504925, −15.85535443320917, −15.23251211096841, −14.28937150242067, −14.08364260586280, −13.86191462586713, −12.85999755265267, −12.49667380444467, −11.89761212865425, −10.92101705098898, −10.09112576511376, −9.914104378463646, −9.309728235179054, −8.607097489542038, −7.988558503480529, −7.197416037003301, −6.404999780605607, −5.509033747036671, −5.197194417539518, −4.536979575202670, −3.523431417207494, −2.636432833412380, −1.740863087348710, −0.7620857873090925, 0.7620857873090925, 1.740863087348710, 2.636432833412380, 3.523431417207494, 4.536979575202670, 5.197194417539518, 5.509033747036671, 6.404999780605607, 7.197416037003301, 7.988558503480529, 8.607097489542038, 9.309728235179054, 9.914104378463646, 10.09112576511376, 10.92101705098898, 11.89761212865425, 12.49667380444467, 12.85999755265267, 13.86191462586713, 14.08364260586280, 14.28937150242067, 15.23251211096841, 15.85535443320917, 16.80861786504925, 17.34862255893244

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