Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·2-s + 2.44·4-s − 0.492·5-s + 7-s − 0.948·8-s + 1.03·10-s + 5.30·13-s − 2.10·14-s − 2.89·16-s − 3.03·17-s − 4.66·19-s − 1.20·20-s + 5.63·23-s − 4.75·25-s − 11.1·26-s + 2.44·28-s − 6.92·29-s − 1.26·31-s + 8.01·32-s + 6.40·34-s − 0.492·35-s + 10.8·37-s + 9.84·38-s + 0.466·40-s + 1.44·41-s + 2.88·43-s − 11.8·46-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.22·4-s − 0.220·5-s + 0.377·7-s − 0.335·8-s + 0.328·10-s + 1.47·13-s − 0.563·14-s − 0.724·16-s − 0.736·17-s − 1.07·19-s − 0.269·20-s + 1.17·23-s − 0.951·25-s − 2.19·26-s + 0.462·28-s − 1.28·29-s − 0.227·31-s + 1.41·32-s + 1.09·34-s − 0.0832·35-s + 1.78·37-s + 1.59·38-s + 0.0738·40-s + 0.225·41-s + 0.439·43-s − 1.75·46-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  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 0.492T + 5T^{2} \)
13 \( 1 - 5.30T + 13T^{2} \)
17 \( 1 + 3.03T + 17T^{2} \)
19 \( 1 + 4.66T + 19T^{2} \)
23 \( 1 - 5.63T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 1.44T + 41T^{2} \)
43 \( 1 - 2.88T + 43T^{2} \)
47 \( 1 - 8.75T + 47T^{2} \)
53 \( 1 + 6.63T + 53T^{2} \)
59 \( 1 - 8.35T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 + 5.94T + 71T^{2} \)
73 \( 1 + 3.77T + 73T^{2} \)
79 \( 1 + 8.80T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + 6.31T + 97T^{2} \)
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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

−7.66004697027783098638778508944, −7.17671638320095793948709939295, −6.26666177764986719414626339153, −5.76273872728203033328489227234, −4.47106105216504100762392272576, −4.03714174429247347846756216014, −2.81889311352473898650731920682, −1.88134794701729154278391089569, −1.13688301700403833279745642239, 0, 1.13688301700403833279745642239, 1.88134794701729154278391089569, 2.81889311352473898650731920682, 4.03714174429247347846756216014, 4.47106105216504100762392272576, 5.76273872728203033328489227234, 6.26666177764986719414626339153, 7.17671638320095793948709939295, 7.66004697027783098638778508944

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