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  + 0.162·2-s − 1.97·4-s + 4.38·5-s + 7-s − 0.644·8-s + 0.711·10-s − 1.67·13-s + 0.162·14-s + 3.84·16-s + 6.18·17-s + 3.47·19-s − 8.65·20-s − 0.568·23-s + 14.2·25-s − 0.271·26-s − 1.97·28-s − 8.86·29-s + 4.33·31-s + 1.91·32-s + 1.00·34-s + 4.38·35-s + 0.969·37-s + 0.563·38-s − 2.82·40-s + 5.77·41-s + 5.04·43-s − 0.0921·46-s + ⋯
L(s)  = 1  + 0.114·2-s − 0.986·4-s + 1.96·5-s + 0.377·7-s − 0.227·8-s + 0.224·10-s − 0.464·13-s + 0.0433·14-s + 0.960·16-s + 1.50·17-s + 0.797·19-s − 1.93·20-s − 0.118·23-s + 2.84·25-s − 0.0532·26-s − 0.372·28-s − 1.64·29-s + 0.778·31-s + 0.337·32-s + 0.172·34-s + 0.741·35-s + 0.159·37-s + 0.0914·38-s − 0.446·40-s + 0.902·41-s + 0.769·43-s − 0.0135·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.962168400$
$L(\frac12)$  $\approx$  $2.962168400$
$L(\frac{3}{2})$   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 \( 1 - 0.162T + 2T^{2} \)
5 \( 1 - 4.38T + 5T^{2} \)
13 \( 1 + 1.67T + 13T^{2} \)
17 \( 1 - 6.18T + 17T^{2} \)
19 \( 1 - 3.47T + 19T^{2} \)
23 \( 1 + 0.568T + 23T^{2} \)
29 \( 1 + 8.86T + 29T^{2} \)
31 \( 1 - 4.33T + 31T^{2} \)
37 \( 1 - 0.969T + 37T^{2} \)
41 \( 1 - 5.77T + 41T^{2} \)
43 \( 1 - 5.04T + 43T^{2} \)
47 \( 1 + 4.67T + 47T^{2} \)
53 \( 1 - 2.37T + 53T^{2} \)
59 \( 1 - 2.84T + 59T^{2} \)
61 \( 1 + 9.29T + 61T^{2} \)
67 \( 1 + 7.14T + 67T^{2} \)
71 \( 1 - 0.794T + 71T^{2} \)
73 \( 1 + 2.38T + 73T^{2} \)
79 \( 1 - 0.670T + 79T^{2} \)
83 \( 1 + 9.28T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−7.83791618651444243586511459499, −7.28853093993346242502165661635, −6.10627552455838696927702989247, −5.72976689926332702283622954120, −5.19200752088708261241976640210, −4.56239202243674935957118415622, −3.46184207845872961317683322444, −2.68089949213873496284019526136, −1.68504486501141907441597385027, −0.921675395505286575241811086068, 0.921675395505286575241811086068, 1.68504486501141907441597385027, 2.68089949213873496284019526136, 3.46184207845872961317683322444, 4.56239202243674935957118415622, 5.19200752088708261241976640210, 5.72976689926332702283622954120, 6.10627552455838696927702989247, 7.28853093993346242502165661635, 7.83791618651444243586511459499

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