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  + 2.68·2-s + 5.18·4-s + 1.17·5-s − 7-s + 8.54·8-s + 3.13·10-s + 1.68·13-s − 2.68·14-s + 12.5·16-s + 5.59·17-s − 4.35·19-s + 6.07·20-s − 0.119·23-s − 3.63·25-s + 4.51·26-s − 5.18·28-s + 4.39·29-s − 6.31·31-s + 16.5·32-s + 15.0·34-s − 1.17·35-s + 5.85·37-s − 11.6·38-s + 9.99·40-s + 11.0·41-s − 5.97·43-s − 0.320·46-s + ⋯
L(s)  = 1  + 1.89·2-s + 2.59·4-s + 0.523·5-s − 0.377·7-s + 3.02·8-s + 0.992·10-s + 0.466·13-s − 0.716·14-s + 3.13·16-s + 1.35·17-s − 0.998·19-s + 1.35·20-s − 0.0249·23-s − 0.726·25-s + 0.884·26-s − 0.980·28-s + 0.816·29-s − 1.13·31-s + 2.91·32-s + 2.57·34-s − 0.197·35-s + 0.962·37-s − 1.89·38-s + 1.58·40-s + 1.72·41-s − 0.910·43-s − 0.0472·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.646119130$
$L(\frac12)$  $\approx$  $8.646119130$
$L(\frac{3}{2})$   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 \( 1 - 2.68T + 2T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
13 \( 1 - 1.68T + 13T^{2} \)
17 \( 1 - 5.59T + 17T^{2} \)
19 \( 1 + 4.35T + 19T^{2} \)
23 \( 1 + 0.119T + 23T^{2} \)
29 \( 1 - 4.39T + 29T^{2} \)
31 \( 1 + 6.31T + 31T^{2} \)
37 \( 1 - 5.85T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 5.97T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 - 0.417T + 53T^{2} \)
59 \( 1 + 0.723T + 59T^{2} \)
61 \( 1 - 8.85T + 61T^{2} \)
67 \( 1 - 8.78T + 67T^{2} \)
71 \( 1 - 3.21T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 3.45T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + 5.99T + 97T^{2} \)
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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

−7.59839159279858919757282823721, −6.82217099414468098974626743077, −6.22404634075541133009176994802, −5.65938798823667370378362548777, −5.22683767288813994737956811711, −4.08154259239251194850859360670, −3.85142153536353053186338461446, −2.81665941914727809256011324495, −2.24790549225065812807308876332, −1.18907238581181833300885785711, 1.18907238581181833300885785711, 2.24790549225065812807308876332, 2.81665941914727809256011324495, 3.85142153536353053186338461446, 4.08154259239251194850859360670, 5.22683767288813994737956811711, 5.65938798823667370378362548777, 6.22404634075541133009176994802, 6.82217099414468098974626743077, 7.59839159279858919757282823721

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