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-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  =  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 \( 1 - T + p T^{2} \)
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

−17.39646409780897, −16.78654388806822, −16.38747482897450, −15.20667288724268, −14.92068645849083, −14.23410694043398, −13.89745246372039, −13.27316081847976, −12.70523223648920, −12.25300149222220, −11.69602477934348, −10.53258118110537, −10.03934806891974, −9.501593640674531, −9.152105041775139, −8.278741275250644, −7.275813011998209, −6.779054830839429, −5.847543195806874, −5.324485128779085, −4.962613105693540, −4.016656100781464, −2.990103402459579, −2.565530429814860, −1.390188046070004, 0, 1.390188046070004, 2.565530429814860, 2.990103402459579, 4.016656100781464, 4.962613105693540, 5.324485128779085, 5.847543195806874, 6.779054830839429, 7.275813011998209, 8.278741275250644, 9.152105041775139, 9.501593640674531, 10.03934806891974, 10.53258118110537, 11.69602477934348, 12.25300149222220, 12.70523223648920, 13.27316081847976, 13.89745246372039, 14.23410694043398, 14.92068645849083, 15.20667288724268, 16.38747482897450, 16.78654388806822, 17.39646409780897

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