Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3231837849$
$L(\frac12)$  $\approx$  $0.3231837849$
$L(\frac{3}{2})$   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

−17.00737029877025, −16.74380593700937, −16.20009375567984, −15.28968146382656, −14.93388766911206, −14.23884610537408, −13.66752184822269, −12.53613250834804, −12.28683401063897, −11.56447286873339, −10.92006744411821, −10.16708868160521, −9.888637395548095, −9.268193929834663, −8.333830397202742, −7.976943931772590, −7.548223841406099, −6.806658123747899, −5.966004662707224, −5.013239405029383, −4.357231963976673, −3.445666687975607, −2.236439561118155, −1.713943935226375, −0.3443769738833848, 0.3443769738833848, 1.713943935226375, 2.236439561118155, 3.445666687975607, 4.357231963976673, 5.013239405029383, 5.966004662707224, 6.806658123747899, 7.548223841406099, 7.976943931772590, 8.333830397202742, 9.268193929834663, 9.888637395548095, 10.16708868160521, 10.92006744411821, 11.56447286873339, 12.28683401063897, 12.53613250834804, 13.66752184822269, 14.23884610537408, 14.93388766911206, 15.28968146382656, 16.20009375567984, 16.74380593700937, 17.00737029877025

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