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  + 0.759·2-s − 1.42·4-s + 2.86·5-s − 7-s − 2.59·8-s + 2.17·10-s + 6.96·13-s − 0.759·14-s + 0.872·16-s − 6.65·17-s + 5.75·19-s − 4.07·20-s − 0.724·23-s + 3.20·25-s + 5.28·26-s + 1.42·28-s − 7.56·29-s + 2.81·31-s + 5.86·32-s − 5.05·34-s − 2.86·35-s − 1.72·37-s + 4.36·38-s − 7.44·40-s − 1.12·41-s + 11.7·43-s − 0.550·46-s + ⋯
L(s)  = 1  + 0.536·2-s − 0.711·4-s + 1.28·5-s − 0.377·7-s − 0.919·8-s + 0.688·10-s + 1.93·13-s − 0.202·14-s + 0.218·16-s − 1.61·17-s + 1.31·19-s − 0.911·20-s − 0.151·23-s + 0.641·25-s + 1.03·26-s + 0.268·28-s − 1.40·29-s + 0.506·31-s + 1.03·32-s − 0.867·34-s − 0.484·35-s − 0.283·37-s + 0.708·38-s − 1.17·40-s − 0.175·41-s + 1.78·43-s − 0.0811·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.777951824$
$L(\frac12)$  $\approx$  $2.777951824$
$L(\frac{3}{2})$   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 \( 1 - 0.759T + 2T^{2} \)
5 \( 1 - 2.86T + 5T^{2} \)
13 \( 1 - 6.96T + 13T^{2} \)
17 \( 1 + 6.65T + 17T^{2} \)
19 \( 1 - 5.75T + 19T^{2} \)
23 \( 1 + 0.724T + 23T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 - 2.81T + 31T^{2} \)
37 \( 1 + 1.72T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 - 5.18T + 47T^{2} \)
53 \( 1 + 6.25T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 - 5.97T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 0.995T + 71T^{2} \)
73 \( 1 + 3.53T + 73T^{2} \)
79 \( 1 - 0.669T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
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

−7.974563071556000674944770175354, −6.91322638074506960002438460332, −6.21243492957374855560077434862, −5.77309956442519455652623251219, −5.24089355961935397552272827522, −4.22159774901263411263963211874, −3.68368789015929938575972025903, −2.80097336264943416579786717798, −1.83788242389732781732054587857, −0.791463000678015970976906404819, 0.791463000678015970976906404819, 1.83788242389732781732054587857, 2.80097336264943416579786717798, 3.68368789015929938575972025903, 4.22159774901263411263963211874, 5.24089355961935397552272827522, 5.77309956442519455652623251219, 6.21243492957374855560077434862, 6.91322638074506960002438460332, 7.974563071556000674944770175354

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