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  + 2·2-s + 2·4-s − 5-s − 7-s − 2·10-s + 6·13-s − 2·14-s − 4·16-s − 7·17-s + 8·19-s − 2·20-s − 6·23-s − 4·25-s + 12·26-s − 2·28-s + 4·29-s + 2·31-s − 8·32-s − 14·34-s + 35-s − 6·37-s + 16·38-s − 2·41-s + 43-s − 12·46-s − 13·47-s + 49-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.66·13-s − 0.534·14-s − 16-s − 1.69·17-s + 1.83·19-s − 0.447·20-s − 1.25·23-s − 4/5·25-s + 2.35·26-s − 0.377·28-s + 0.742·29-s + 0.359·31-s − 1.41·32-s − 2.40·34-s + 0.169·35-s − 0.986·37-s + 2.59·38-s − 0.312·41-s + 0.152·43-s − 1.76·46-s − 1.89·47-s + 1/7·49-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
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 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−17.47577366650288, −16.18092086238188, −15.90142836383481, −15.74145415764834, −15.04093688641965, −14.11180993775355, −13.78931662674903, −13.35494451260672, −12.82351009024842, −11.97391660396968, −11.57337741458385, −11.18841763730506, −10.24870496157623, −9.545570470378048, −8.711813862731194, −8.246514960866144, −7.258849910762570, −6.535345999686368, −6.100124915076804, −5.367429279465222, −4.594951071076725, −3.861744280778836, −3.470136203059299, −2.612899088460155, −1.510679785876709, 0, 1.510679785876709, 2.612899088460155, 3.470136203059299, 3.861744280778836, 4.594951071076725, 5.367429279465222, 6.100124915076804, 6.535345999686368, 7.258849910762570, 8.246514960866144, 8.711813862731194, 9.545570470378048, 10.24870496157623, 11.18841763730506, 11.57337741458385, 11.97391660396968, 12.82351009024842, 13.35494451260672, 13.78931662674903, 14.11180993775355, 15.04093688641965, 15.74145415764834, 15.90142836383481, 16.18092086238188, 17.47577366650288

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