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  + 0.473·2-s − 1.77·4-s − 3.75·5-s + 7-s − 1.78·8-s − 1.78·10-s − 2.70·13-s + 0.473·14-s + 2.70·16-s + 2.48·17-s + 1.74·19-s + 6.67·20-s − 3.79·23-s + 9.12·25-s − 1.28·26-s − 1.77·28-s − 5.80·29-s + 2.24·31-s + 4.85·32-s + 1.17·34-s − 3.75·35-s − 7.65·37-s + 0.826·38-s + 6.72·40-s − 4.18·41-s − 7.75·43-s − 1.79·46-s + ⋯
L(s)  = 1  + 0.334·2-s − 0.887·4-s − 1.68·5-s + 0.377·7-s − 0.632·8-s − 0.563·10-s − 0.750·13-s + 0.126·14-s + 0.675·16-s + 0.602·17-s + 0.400·19-s + 1.49·20-s − 0.792·23-s + 1.82·25-s − 0.251·26-s − 0.335·28-s − 1.07·29-s + 0.403·31-s + 0.858·32-s + 0.201·34-s − 0.635·35-s − 1.25·37-s + 0.134·38-s + 1.06·40-s − 0.653·41-s − 1.18·43-s − 0.265·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5154689911$
$L(\frac12)$  $\approx$  $0.5154689911$
$L(\frac{3}{2})$   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 \( 1 - 0.473T + 2T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
13 \( 1 + 2.70T + 13T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 - 1.74T + 19T^{2} \)
23 \( 1 + 3.79T + 23T^{2} \)
29 \( 1 + 5.80T + 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 4.18T + 41T^{2} \)
43 \( 1 + 7.75T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 8.83T + 53T^{2} \)
59 \( 1 + 4.54T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 7.05T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 - 6.99T + 97T^{2} \)
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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

−7.82515418302844249596193200578, −7.46252836269000117305399411014, −6.53465704927278254566553301206, −5.52489459987720916952163510312, −4.83738508243455063899341739184, −4.42903539436571908517550665317, −3.44922330681006592402191341817, −3.26084888123219521460214528746, −1.68407714579041391185556476754, −0.34266297581859492527457302155, 0.34266297581859492527457302155, 1.68407714579041391185556476754, 3.26084888123219521460214528746, 3.44922330681006592402191341817, 4.42903539436571908517550665317, 4.83738508243455063899341739184, 5.52489459987720916952163510312, 6.53465704927278254566553301206, 7.46252836269000117305399411014, 7.82515418302844249596193200578

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