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  − 2.73·2-s + 5.46·4-s + 3.73·5-s − 7-s − 9.46·8-s − 10.1·10-s − 0.732·13-s + 2.73·14-s + 14.9·16-s + 0.267·17-s + 8.19·19-s + 20.3·20-s − 6.73·23-s + 8.92·25-s + 2·26-s − 5.46·28-s − 4.73·29-s + 0.535·31-s − 21.8·32-s − 0.732·34-s − 3.73·35-s − 2.53·37-s − 22.3·38-s − 35.3·40-s − 4·41-s − 6.46·43-s + 18.3·46-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.73·4-s + 1.66·5-s − 0.377·7-s − 3.34·8-s − 3.22·10-s − 0.203·13-s + 0.730·14-s + 3.73·16-s + 0.0649·17-s + 1.88·19-s + 4.55·20-s − 1.40·23-s + 1.78·25-s + 0.392·26-s − 1.03·28-s − 0.878·29-s + 0.0962·31-s − 3.86·32-s − 0.125·34-s − 0.630·35-s − 0.416·37-s − 3.63·38-s − 5.58·40-s − 0.624·41-s − 0.985·43-s + 2.71·46-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 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
13 \( 1 + 0.732T + 13T^{2} \)
17 \( 1 - 0.267T + 17T^{2} \)
19 \( 1 - 8.19T + 19T^{2} \)
23 \( 1 + 6.73T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 - 0.535T + 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6.46T + 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + 9.46T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + 3.26T + 73T^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + 7.73T + 83T^{2} \)
89 \( 1 + 2.66T + 89T^{2} \)
97 \( 1 - 6.73T + 97T^{2} \)
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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

−7.63904652206657779081458307581, −7.00975015173377592437661304972, −6.28061646112836672212387731178, −5.83302423305163685350634578220, −5.10373080308573760173316594488, −3.39543698488040164789938736829, −2.72284744247203233236987706637, −1.80965353932400059676799728791, −1.36079094486045536322818369327, 0, 1.36079094486045536322818369327, 1.80965353932400059676799728791, 2.72284744247203233236987706637, 3.39543698488040164789938736829, 5.10373080308573760173316594488, 5.83302423305163685350634578220, 6.28061646112836672212387731178, 7.00975015173377592437661304972, 7.63904652206657779081458307581

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