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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.250173744$
$L(\frac12)$  $\approx$  $1.250173744$
$L(\frac{3}{2})$   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

−7.919675326190093141823780157148, −7.53802172656293307443679918558, −6.60912249065091641673443073495, −5.74128358797813518039843863885, −5.05315721824688294242691410975, −4.06276914903436120197385029369, −3.65573345300378246706318540162, −2.98841612874647809966665611063, −1.17188673130087490301791007906, −0.78297010862927210340556488729, 0.78297010862927210340556488729, 1.17188673130087490301791007906, 2.98841612874647809966665611063, 3.65573345300378246706318540162, 4.06276914903436120197385029369, 5.05315721824688294242691410975, 5.74128358797813518039843863885, 6.60912249065091641673443073495, 7.53802172656293307443679918558, 7.919675326190093141823780157148

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