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  + 0.470·2-s − 1.77·4-s − 3.24·5-s − 7-s − 1.77·8-s − 1.52·10-s − 2.47·13-s − 0.470·14-s + 2.71·16-s + 3.24·17-s − 5.30·19-s + 5.77·20-s + 8.86·23-s + 5.55·25-s − 1.16·26-s + 1.77·28-s + 2.47·29-s − 6.49·31-s + 4.83·32-s + 1.52·34-s + 3.24·35-s + 1.77·37-s − 2.49·38-s + 5.77·40-s + 1.28·41-s − 4.89·43-s + 4.17·46-s + ⋯
L(s)  = 1  + 0.332·2-s − 0.889·4-s − 1.45·5-s − 0.377·7-s − 0.628·8-s − 0.483·10-s − 0.685·13-s − 0.125·14-s + 0.679·16-s + 0.788·17-s − 1.21·19-s + 1.29·20-s + 1.84·23-s + 1.11·25-s − 0.228·26-s + 0.336·28-s + 0.458·29-s − 1.16·31-s + 0.855·32-s + 0.262·34-s + 0.549·35-s + 0.292·37-s − 0.405·38-s + 0.913·40-s + 0.199·41-s − 0.746·43-s + 0.615·46-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 \( 1 - 0.470T + 2T^{2} \)
5 \( 1 + 3.24T + 5T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 - 3.24T + 17T^{2} \)
19 \( 1 + 5.30T + 19T^{2} \)
23 \( 1 - 8.86T + 23T^{2} \)
29 \( 1 - 2.47T + 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 + 4.89T + 43T^{2} \)
47 \( 1 + 3.96T + 47T^{2} \)
53 \( 1 - 9.77T + 53T^{2} \)
59 \( 1 + 2.41T + 59T^{2} \)
61 \( 1 - 4.74T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 - 3.19T + 71T^{2} \)
73 \( 1 - 8.98T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 7.86T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−7.44925753804600215320038442780, −7.02018855868030406447974577884, −6.08730902441472551775736122778, −5.10661737654976694723737557982, −4.74493085311436246762915935773, −3.80481081887071681648686194081, −3.48287112928752498609953364640, −2.53206740589611133411818548199, −0.906686163267401990341140566352, 0, 0.906686163267401990341140566352, 2.53206740589611133411818548199, 3.48287112928752498609953364640, 3.80481081887071681648686194081, 4.74493085311436246762915935773, 5.10661737654976694723737557982, 6.08730902441472551775736122778, 7.02018855868030406447974577884, 7.44925753804600215320038442780

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