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  + 1.31·2-s − 0.276·4-s + 3.45·5-s + 7-s − 2.98·8-s + 4.54·10-s − 5.53·13-s + 1.31·14-s − 3.37·16-s − 4.96·17-s + 8.22·19-s − 0.955·20-s + 7.56·23-s + 6.96·25-s − 7.27·26-s − 0.276·28-s + 1.47·29-s + 2.41·31-s + 1.55·32-s − 6.51·34-s + 3.45·35-s − 2.83·37-s + 10.8·38-s − 10.3·40-s − 6.93·41-s + 5.62·43-s + 9.92·46-s + ⋯
L(s)  = 1  + 0.928·2-s − 0.138·4-s + 1.54·5-s + 0.377·7-s − 1.05·8-s + 1.43·10-s − 1.53·13-s + 0.350·14-s − 0.842·16-s − 1.20·17-s + 1.88·19-s − 0.213·20-s + 1.57·23-s + 1.39·25-s − 1.42·26-s − 0.0522·28-s + 0.272·29-s + 0.433·31-s + 0.274·32-s − 1.11·34-s + 0.584·35-s − 0.466·37-s + 1.75·38-s − 1.63·40-s − 1.08·41-s + 0.858·43-s + 1.46·46-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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.856580592$
$L(\frac12)$  $\approx$  $3.856580592$
$L(\frac{3}{2})$   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 \( 1 - 1.31T + 2T^{2} \)
5 \( 1 - 3.45T + 5T^{2} \)
13 \( 1 + 5.53T + 13T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 - 8.22T + 19T^{2} \)
23 \( 1 - 7.56T + 23T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 - 2.41T + 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + 6.93T + 41T^{2} \)
43 \( 1 - 5.62T + 43T^{2} \)
47 \( 1 - 0.0417T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 2.99T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 - 7.38T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 - 9.64T + 83T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + 2.35T + 97T^{2} \)
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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

−7.72432377077106873309387346744, −6.90388715942116925619519911566, −6.40447241174378064718184888757, −5.44599300010362545038402617964, −5.05269227665946907480851431493, −4.73183419283297089051360801708, −3.49236968408782930695756727106, −2.68989747013591375474099015020, −2.11426752741926973947386698539, −0.857665288104054720301863068936, 0.857665288104054720301863068936, 2.11426752741926973947386698539, 2.68989747013591375474099015020, 3.49236968408782930695756727106, 4.73183419283297089051360801708, 5.05269227665946907480851431493, 5.44599300010362545038402617964, 6.40447241174378064718184888757, 6.90388715942116925619519911566, 7.72432377077106873309387346744

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