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  − 2.10·2-s + 2.42·4-s + 3.75·5-s − 7-s − 0.897·8-s − 7.89·10-s − 3.46·13-s + 2.10·14-s − 2.96·16-s + 1.52·17-s + 5.16·19-s + 9.10·20-s + 4.87·23-s + 9.06·25-s + 7.29·26-s − 2.42·28-s + 10.7·29-s + 8.12·31-s + 8.03·32-s − 3.21·34-s − 3.75·35-s + 9.25·37-s − 10.8·38-s − 3.36·40-s − 10.8·41-s + 0.137·43-s − 10.2·46-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.21·4-s + 1.67·5-s − 0.377·7-s − 0.317·8-s − 2.49·10-s − 0.961·13-s + 0.562·14-s − 0.741·16-s + 0.370·17-s + 1.18·19-s + 2.03·20-s + 1.01·23-s + 1.81·25-s + 1.43·26-s − 0.458·28-s + 1.99·29-s + 1.45·31-s + 1.42·32-s − 0.550·34-s − 0.633·35-s + 1.52·37-s − 1.76·38-s − 0.532·40-s − 1.70·41-s + 0.0209·43-s − 1.51·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.449816132$
$L(\frac12)$  $\approx$  $1.449816132$
$L(\frac{3}{2})$   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 \( 1 + 2.10T + 2T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 1.52T + 17T^{2} \)
19 \( 1 - 5.16T + 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 - 10.7T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 - 9.25T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 0.137T + 43T^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 - 2.79T + 53T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 - 0.297T + 61T^{2} \)
67 \( 1 - 0.816T + 67T^{2} \)
71 \( 1 + 5.72T + 71T^{2} \)
73 \( 1 - 2.47T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−8.097909996388331515536037119348, −7.10944107257569229237301752364, −6.77757863695929336904214217415, −5.99141362629681378552292947294, −5.19522227534707019318264988005, −4.55600928820952417737234130703, −2.85344008776422560422090306992, −2.62467882778233917721345490865, −1.44874439686589303564898207505, −0.835122253052977125304300667495, 0.835122253052977125304300667495, 1.44874439686589303564898207505, 2.62467882778233917721345490865, 2.85344008776422560422090306992, 4.55600928820952417737234130703, 5.19522227534707019318264988005, 5.99141362629681378552292947294, 6.77757863695929336904214217415, 7.10944107257569229237301752364, 8.097909996388331515536037119348

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