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  − 2·4-s − 3.60·5-s + 7-s − 2·13-s + 4·16-s + 3.60·17-s − 6·19-s + 7.21·20-s − 7.21·23-s + 7.99·25-s − 2·28-s + 7.21·29-s − 2·31-s − 3.60·35-s − 2·37-s + 7.21·41-s + 5·43-s − 3.60·47-s + 49-s + 4·52-s + 10.8·59-s + 14·61-s − 8·64-s + 7.21·65-s − 15·67-s − 7.21·68-s + 14.4·71-s + ⋯
L(s)  = 1  − 4-s − 1.61·5-s + 0.377·7-s − 0.554·13-s + 16-s + 0.874·17-s − 1.37·19-s + 1.61·20-s − 1.50·23-s + 1.59·25-s − 0.377·28-s + 1.33·29-s − 0.359·31-s − 0.609·35-s − 0.328·37-s + 1.12·41-s + 0.762·43-s − 0.525·47-s + 0.142·49-s + 0.554·52-s + 1.40·59-s + 1.79·61-s − 64-s + 0.894·65-s − 1.83·67-s − 0.874·68-s + 1.71·71-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  :  $\chi_{7623} (1, \cdot )$
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 \( 1 + 2T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 3.60T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 7.21T + 23T^{2} \)
29 \( 1 - 7.21T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 7.21T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + 3.60T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 15T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−7.74548315788270244442390894623, −7.04983755009916007187010712185, −6.07649167917037921142061042238, −5.24036065205136004085315406640, −4.45131399783077334962623494481, −4.09141912673313470117227320076, −3.42165639423045718245830645924, −2.32164320616891640454317839390, −0.904918320736879303599730197489, 0, 0.904918320736879303599730197489, 2.32164320616891640454317839390, 3.42165639423045718245830645924, 4.09141912673313470117227320076, 4.45131399783077334962623494481, 5.24036065205136004085315406640, 6.07649167917037921142061042238, 7.04983755009916007187010712185, 7.74548315788270244442390894623

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