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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7138495111$
$L(\frac12)$  $\approx$  $0.7138495111$
$L(\frac{3}{2})$   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

−7.932239396113975755640796128333, −7.33305129302908929280077061113, −6.88961079138409812535166729392, −5.96881123436764030228157277874, −5.43998676528858655144039606242, −4.25527758432103998143568829469, −3.09345949447602338429569988076, −2.17631374487704363857296398425, −1.72629358672953729158652966838, −0.55237059145286590960719476806, 0.55237059145286590960719476806, 1.72629358672953729158652966838, 2.17631374487704363857296398425, 3.09345949447602338429569988076, 4.25527758432103998143568829469, 5.43998676528858655144039606242, 5.96881123436764030228157277874, 6.88961079138409812535166729392, 7.33305129302908929280077061113, 7.932239396113975755640796128333

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