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  + 2.47·2-s + 4.11·4-s + 2.58·5-s + 7-s + 5.22·8-s + 6.39·10-s + 5.87·13-s + 2.47·14-s + 4.70·16-s + 7.51·17-s + 2.35·19-s + 10.6·20-s − 6.94·23-s + 1.69·25-s + 14.5·26-s + 4.11·28-s − 5.87·29-s − 3.66·31-s + 1.16·32-s + 18.5·34-s + 2.58·35-s + 3.30·37-s + 5.83·38-s + 13.5·40-s + 5.28·41-s − 7.40·43-s − 17.1·46-s + ⋯
L(s)  = 1  + 1.74·2-s + 2.05·4-s + 1.15·5-s + 0.377·7-s + 1.84·8-s + 2.02·10-s + 1.62·13-s + 0.660·14-s + 1.17·16-s + 1.82·17-s + 0.540·19-s + 2.38·20-s − 1.44·23-s + 0.339·25-s + 2.84·26-s + 0.777·28-s − 1.09·29-s − 0.657·31-s + 0.206·32-s + 3.18·34-s + 0.437·35-s + 0.543·37-s + 0.945·38-s + 2.13·40-s + 0.825·41-s − 1.12·43-s − 2.53·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.265284299$
$L(\frac12)$  $\approx$  $9.265284299$
$L(\frac{3}{2})$   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 \( 1 - 2.47T + 2T^{2} \)
5 \( 1 - 2.58T + 5T^{2} \)
13 \( 1 - 5.87T + 13T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 + 6.94T + 23T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + 3.66T + 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 + 7.40T + 43T^{2} \)
47 \( 1 + 7.53T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 - 0.926T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.69472291003485105731155528866, −6.86053027695925187885525982916, −5.93162407070631273236998177169, −5.80521282517217095017927290241, −5.30111751047685863997206755598, −4.25109106993133927387803178021, −3.62485896320155392971296729831, −2.99228654683653474208806287522, −1.89872188842141517084720469470, −1.38966823335737992238781390990, 1.38966823335737992238781390990, 1.89872188842141517084720469470, 2.99228654683653474208806287522, 3.62485896320155392971296729831, 4.25109106993133927387803178021, 5.30111751047685863997206755598, 5.80521282517217095017927290241, 5.93162407070631273236998177169, 6.86053027695925187885525982916, 7.69472291003485105731155528866

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