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  − 2.09·2-s + 2.40·4-s − 3.15·5-s + 7-s − 0.839·8-s + 6.62·10-s − 2.12·13-s − 2.09·14-s − 3.03·16-s − 4.79·17-s − 1.53·19-s − 7.58·20-s − 5.11·23-s + 4.98·25-s + 4.46·26-s + 2.40·28-s − 0.958·29-s − 7.22·31-s + 8.05·32-s + 10.0·34-s − 3.15·35-s + 2.39·37-s + 3.22·38-s + 2.65·40-s + 0.266·41-s − 9.28·43-s + 10.7·46-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.20·4-s − 1.41·5-s + 0.377·7-s − 0.296·8-s + 2.09·10-s − 0.590·13-s − 0.560·14-s − 0.759·16-s − 1.16·17-s − 0.352·19-s − 1.69·20-s − 1.06·23-s + 0.996·25-s + 0.875·26-s + 0.453·28-s − 0.177·29-s − 1.29·31-s + 1.42·32-s + 1.72·34-s − 0.534·35-s + 0.394·37-s + 0.523·38-s + 0.419·40-s + 0.0416·41-s − 1.41·43-s + 1.58·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1045499162$
$L(\frac12)$  $\approx$  $0.1045499162$
$L(\frac{3}{2})$   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 \( 1 + 2.09T + 2T^{2} \)
5 \( 1 + 3.15T + 5T^{2} \)
13 \( 1 + 2.12T + 13T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 + 5.11T + 23T^{2} \)
29 \( 1 + 0.958T + 29T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 - 0.266T + 41T^{2} \)
43 \( 1 + 9.28T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 0.945T + 53T^{2} \)
59 \( 1 - 9.55T + 59T^{2} \)
61 \( 1 + 8.55T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 7.72T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 7.08T + 97T^{2} \)
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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

−8.052585580255016254680980648130, −7.33065773600037837096545101523, −7.00293974953394356974716745463, −6.06135072203614745091206671211, −4.85848179932135332698510543851, −4.32689181402676226403454395045, −3.51163950339288876764750915516, −2.35138274430449711104991069799, −1.57990259056220964155235500442, −0.20166689580682992814219565898, 0.20166689580682992814219565898, 1.57990259056220964155235500442, 2.35138274430449711104991069799, 3.51163950339288876764750915516, 4.32689181402676226403454395045, 4.85848179932135332698510543851, 6.06135072203614745091206671211, 7.00293974953394356974716745463, 7.33065773600037837096545101523, 8.052585580255016254680980648130

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