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  − 1.60·2-s + 0.568·4-s − 1.76·5-s − 7-s + 2.29·8-s + 2.82·10-s + 0.237·13-s + 1.60·14-s − 4.81·16-s + 6.14·17-s + 1.53·19-s − 1.00·20-s − 3.89·23-s − 1.88·25-s − 0.381·26-s − 0.568·28-s − 4.45·29-s + 8.52·31-s + 3.12·32-s − 9.84·34-s + 1.76·35-s − 3.64·37-s − 2.46·38-s − 4.04·40-s − 1.90·41-s − 11.7·43-s + 6.24·46-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.284·4-s − 0.789·5-s − 0.377·7-s + 0.811·8-s + 0.894·10-s + 0.0659·13-s + 0.428·14-s − 1.20·16-s + 1.48·17-s + 0.352·19-s − 0.224·20-s − 0.812·23-s − 0.377·25-s − 0.0747·26-s − 0.107·28-s − 0.827·29-s + 1.53·31-s + 0.552·32-s − 1.68·34-s + 0.298·35-s − 0.599·37-s − 0.399·38-s − 0.640·40-s − 0.297·41-s − 1.78·43-s + 0.920·46-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 \( 1 + 1.60T + 2T^{2} \)
5 \( 1 + 1.76T + 5T^{2} \)
13 \( 1 - 0.237T + 13T^{2} \)
17 \( 1 - 6.14T + 17T^{2} \)
19 \( 1 - 1.53T + 19T^{2} \)
23 \( 1 + 3.89T + 23T^{2} \)
29 \( 1 + 4.45T + 29T^{2} \)
31 \( 1 - 8.52T + 31T^{2} \)
37 \( 1 + 3.64T + 37T^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 8.55T + 47T^{2} \)
53 \( 1 + 3.14T + 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 + 6.78T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 - 7.37T + 71T^{2} \)
73 \( 1 - 4.60T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 - 1.14T + 89T^{2} \)
97 \( 1 + 4.68T + 97T^{2} \)
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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

−7.71345323941883064827335986472, −7.19255136147686045910148354198, −6.31601291653004253225558710299, −5.47693323004495475872141450079, −4.63880557609886518601786689580, −3.80762989853085136835631891393, −3.18398942524112759144165677929, −1.94844355894843665033444127406, −0.974535851661545281747643310737, 0, 0.974535851661545281747643310737, 1.94844355894843665033444127406, 3.18398942524112759144165677929, 3.80762989853085136835631891393, 4.63880557609886518601786689580, 5.47693323004495475872141450079, 6.31601291653004253225558710299, 7.19255136147686045910148354198, 7.71345323941883064827335986472

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