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  − 0.618·2-s − 1.61·4-s − 5-s − 7-s + 2.23·8-s + 0.618·10-s − 3.47·13-s + 0.618·14-s + 1.85·16-s + 5.23·17-s + 6.70·19-s + 1.61·20-s − 5.70·23-s − 4·25-s + 2.14·26-s + 1.61·28-s + 5·29-s − 5.23·31-s − 5.61·32-s − 3.23·34-s + 35-s − 7·37-s − 4.14·38-s − 2.23·40-s − 2.47·41-s − 5.70·43-s + 3.52·46-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 0.447·5-s − 0.377·7-s + 0.790·8-s + 0.195·10-s − 0.962·13-s + 0.165·14-s + 0.463·16-s + 1.26·17-s + 1.53·19-s + 0.361·20-s − 1.19·23-s − 0.800·25-s + 0.420·26-s + 0.305·28-s + 0.928·29-s − 0.940·31-s − 0.993·32-s − 0.554·34-s + 0.169·35-s − 1.15·37-s − 0.672·38-s − 0.353·40-s − 0.386·41-s − 0.870·43-s + 0.520·46-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 \( 1 + 0.618T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
13 \( 1 + 3.47T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + 0.236T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 + 4.52T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 6.76T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 9.70T + 97T^{2} \)
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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

−7.50681152914112853247300145288, −7.25545466232050191809061493116, −6.04751281375309092309197997169, −5.30193137437481278032074178052, −4.82386464855389994609493442070, −3.69881020298103924638242649404, −3.43304295095258331453813440752, −2.13247712610977997763890786436, −0.995850504174931781794216194055, 0, 0.995850504174931781794216194055, 2.13247712610977997763890786436, 3.43304295095258331453813440752, 3.69881020298103924638242649404, 4.82386464855389994609493442070, 5.30193137437481278032074178052, 6.04751281375309092309197997169, 7.25545466232050191809061493116, 7.50681152914112853247300145288

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