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  + 1.41·2-s − 2.64·5-s + 7-s − 2.82·8-s − 3.74·10-s − 5.74·13-s + 1.41·14-s − 4.00·16-s + 5.47·17-s + 5.74·19-s + 3.87·23-s + 2.00·25-s − 8.11·26-s + 3.87·29-s + 5.48·31-s + 7.74·34-s − 2.64·35-s + 3.48·37-s + 8.11·38-s + 7.48·40-s − 5.65·41-s − 43-s + 5.48·46-s + 5.47·47-s + 49-s + ⋯
L(s)  = 1  + 1.00·2-s − 1.18·5-s + 0.377·7-s − 0.999·8-s − 1.18·10-s − 1.59·13-s + 0.377·14-s − 1.00·16-s + 1.32·17-s + 1.31·19-s + 0.808·23-s + 0.400·25-s − 1.59·26-s + 0.719·29-s + 0.984·31-s + 1.32·34-s − 0.447·35-s + 0.572·37-s + 1.31·38-s + 1.18·40-s − 0.883·41-s − 0.152·43-s + 0.808·46-s + 0.798·47-s + 0.142·49-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 \( 1 - 1.41T + 2T^{2} \)
5 \( 1 + 2.64T + 5T^{2} \)
13 \( 1 + 5.74T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 - 5.74T + 19T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 - 5.48T + 31T^{2} \)
37 \( 1 - 3.48T + 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 5.47T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 8.30T + 59T^{2} \)
61 \( 1 - 1.74T + 61T^{2} \)
67 \( 1 + 6.48T + 67T^{2} \)
71 \( 1 - 1.41T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 5.83T + 89T^{2} \)
97 \( 1 + 7.22T + 97T^{2} \)
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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

−7.57011161749605087307977108989, −6.89408848161615281528888403887, −5.90194548175669496671433893437, −5.12449460439855861930337251663, −4.74262442628845806878341727539, −4.07535672915146882861037143408, −3.09412214047482753285799214724, −2.85768337670701013958104659881, −1.19189014388242068210239139133, 0, 1.19189014388242068210239139133, 2.85768337670701013958104659881, 3.09412214047482753285799214724, 4.07535672915146882861037143408, 4.74262442628845806878341727539, 5.12449460439855861930337251663, 5.90194548175669496671433893437, 6.89408848161615281528888403887, 7.57011161749605087307977108989

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