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  + 2.79·2-s + 5.83·4-s + 1.67·5-s + 7-s + 10.7·8-s + 4.69·10-s − 5.87·13-s + 2.79·14-s + 18.3·16-s − 0.304·17-s + 4.62·19-s + 9.78·20-s + 1.38·23-s − 2.18·25-s − 16.4·26-s + 5.83·28-s + 3.96·29-s + 0.764·31-s + 29.9·32-s − 0.851·34-s + 1.67·35-s + 3.21·37-s + 12.9·38-s + 17.9·40-s − 8.31·41-s + 1.01·43-s + 3.86·46-s + ⋯
L(s)  = 1  + 1.97·2-s + 2.91·4-s + 0.750·5-s + 0.377·7-s + 3.79·8-s + 1.48·10-s − 1.62·13-s + 0.747·14-s + 4.58·16-s − 0.0737·17-s + 1.06·19-s + 2.18·20-s + 0.288·23-s − 0.437·25-s − 3.22·26-s + 1.10·28-s + 0.736·29-s + 0.137·31-s + 5.28·32-s − 0.145·34-s + 0.283·35-s + 0.527·37-s + 2.10·38-s + 2.84·40-s − 1.29·41-s + 0.154·43-s + 0.570·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $10.11952178$
$L(\frac12)$  $\approx$  $10.11952178$
$L(\frac{3}{2})$   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 \( 1 - 2.79T + 2T^{2} \)
5 \( 1 - 1.67T + 5T^{2} \)
13 \( 1 + 5.87T + 13T^{2} \)
17 \( 1 + 0.304T + 17T^{2} \)
19 \( 1 - 4.62T + 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 - 0.764T + 31T^{2} \)
37 \( 1 - 3.21T + 37T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 - 1.01T + 43T^{2} \)
47 \( 1 - 2.58T + 47T^{2} \)
53 \( 1 - 1.45T + 53T^{2} \)
59 \( 1 - 2.49T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 - 6.16T + 67T^{2} \)
71 \( 1 + 7.59T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + 2.01T + 83T^{2} \)
89 \( 1 + 0.0843T + 89T^{2} \)
97 \( 1 - 1.03T + 97T^{2} \)
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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

−7.51046887462366274618317229954, −6.94155189960058177287426470392, −6.30491392201370318504405603910, −5.45692416506931809435723889402, −5.12588595853851112749894627727, −4.50691145674775076901520397183, −3.63013546813850010312424191998, −2.73004268311064683472228196863, −2.26600938885505864241736041076, −1.31511371623351703495988499243, 1.31511371623351703495988499243, 2.26600938885505864241736041076, 2.73004268311064683472228196863, 3.63013546813850010312424191998, 4.50691145674775076901520397183, 5.12588595853851112749894627727, 5.45692416506931809435723889402, 6.30491392201370318504405603910, 6.94155189960058177287426470392, 7.51046887462366274618317229954

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