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  + 0.732·2-s − 1.46·4-s + 0.267·5-s − 7-s − 2.53·8-s + 0.196·10-s + 2.73·13-s − 0.732·14-s + 1.07·16-s + 3.73·17-s − 2.19·19-s − 0.392·20-s − 3.26·23-s − 4.92·25-s + 2·26-s + 1.46·28-s − 1.26·29-s + 7.46·31-s + 5.85·32-s + 2.73·34-s − 0.267·35-s − 9.46·37-s − 1.60·38-s − 0.679·40-s − 4·41-s + 0.464·43-s − 2.39·46-s + ⋯
L(s)  = 1  + 0.517·2-s − 0.732·4-s + 0.119·5-s − 0.377·7-s − 0.896·8-s + 0.0620·10-s + 0.757·13-s − 0.195·14-s + 0.267·16-s + 0.905·17-s − 0.503·19-s − 0.0877·20-s − 0.681·23-s − 0.985·25-s + 0.392·26-s + 0.276·28-s − 0.235·29-s + 1.34·31-s + 1.03·32-s + 0.468·34-s − 0.0452·35-s − 1.55·37-s − 0.260·38-s − 0.107·40-s − 0.624·41-s + 0.0707·43-s − 0.352·46-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 \( 1 - 0.732T + 2T^{2} \)
5 \( 1 - 0.267T + 5T^{2} \)
13 \( 1 - 2.73T + 13T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 + 1.26T + 29T^{2} \)
31 \( 1 - 7.46T + 31T^{2} \)
37 \( 1 + 9.46T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 0.464T + 43T^{2} \)
47 \( 1 - 9.19T + 47T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 8.19T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 9.12T + 71T^{2} \)
73 \( 1 + 6.73T + 73T^{2} \)
79 \( 1 + 0.535T + 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−7.56281493288620956313322006548, −6.64698133836767640949023354429, −5.92805958479976010042178001096, −5.53339743087358398309661294709, −4.63266477778450701817866466001, −3.86387575841633374180560709588, −3.41902814943474054848937196136, −2.40510663050330155554048137452, −1.21064718299637104702911621695, 0, 1.21064718299637104702911621695, 2.40510663050330155554048137452, 3.41902814943474054848937196136, 3.86387575841633374180560709588, 4.63266477778450701817866466001, 5.53339743087358398309661294709, 5.92805958479976010042178001096, 6.64698133836767640949023354429, 7.56281493288620956313322006548

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