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  − 2.79·2-s + 5.80·4-s + 1.62·5-s − 7-s − 10.6·8-s − 4.54·10-s − 6.23·13-s + 2.79·14-s + 18.1·16-s + 6.00·17-s + 0.305·19-s + 9.44·20-s + 7.84·23-s − 2.35·25-s + 17.4·26-s − 5.80·28-s + 2.94·29-s − 5.31·31-s − 29.3·32-s − 16.7·34-s − 1.62·35-s − 6.41·37-s − 0.854·38-s − 17.2·40-s + 4.04·41-s − 0.640·43-s − 21.9·46-s + ⋯
L(s)  = 1  − 1.97·2-s + 2.90·4-s + 0.727·5-s − 0.377·7-s − 3.76·8-s − 1.43·10-s − 1.72·13-s + 0.746·14-s + 4.52·16-s + 1.45·17-s + 0.0701·19-s + 2.11·20-s + 1.63·23-s − 0.470·25-s + 3.41·26-s − 1.09·28-s + 0.547·29-s − 0.954·31-s − 5.18·32-s − 2.87·34-s − 0.274·35-s − 1.05·37-s − 0.138·38-s − 2.73·40-s + 0.631·41-s − 0.0977·43-s − 3.23·46-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 \( 1 + 2.79T + 2T^{2} \)
5 \( 1 - 1.62T + 5T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 - 0.305T + 19T^{2} \)
23 \( 1 - 7.84T + 23T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + 5.31T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 + 0.640T + 43T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + 0.251T + 53T^{2} \)
59 \( 1 + 5.46T + 59T^{2} \)
61 \( 1 - 2.45T + 61T^{2} \)
67 \( 1 - 6.54T + 67T^{2} \)
71 \( 1 + 5.61T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 9.59T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−7.55457559137588361505602172688, −7.14036306137699224407185915669, −6.48989195163261627443035538354, −5.65076592025465747545690366494, −5.07709679000561605253096640700, −3.37350170030467518841021723439, −2.76851930866080831612060269681, −1.97223858083670739913636496463, −1.11679647210910848847303413702, 0, 1.11679647210910848847303413702, 1.97223858083670739913636496463, 2.76851930866080831612060269681, 3.37350170030467518841021723439, 5.07709679000561605253096640700, 5.65076592025465747545690366494, 6.48989195163261627443035538354, 7.14036306137699224407185915669, 7.55457559137588361505602172688

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