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  + 2.43·2-s + 3.90·4-s − 1.43·5-s + 7-s + 4.64·8-s − 3.47·10-s − 2.30·13-s + 2.43·14-s + 3.46·16-s + 1.14·17-s − 5.47·19-s − 5.59·20-s − 3.00·23-s − 2.95·25-s − 5.59·26-s + 3.90·28-s − 3.16·29-s − 6.99·31-s − 0.861·32-s + 2.78·34-s − 1.43·35-s − 8.16·37-s − 13.3·38-s − 6.64·40-s + 5.59·41-s − 10.5·43-s − 7.30·46-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.95·4-s − 0.639·5-s + 0.377·7-s + 1.64·8-s − 1.09·10-s − 0.638·13-s + 0.649·14-s + 0.866·16-s + 0.277·17-s − 1.25·19-s − 1.25·20-s − 0.626·23-s − 0.590·25-s − 1.09·26-s + 0.738·28-s − 0.588·29-s − 1.25·31-s − 0.152·32-s + 0.476·34-s − 0.241·35-s − 1.34·37-s − 2.16·38-s − 1.05·40-s + 0.873·41-s − 1.61·43-s − 1.07·46-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 \( 1 - 2.43T + 2T^{2} \)
5 \( 1 + 1.43T + 5T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
23 \( 1 + 3.00T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 6.99T + 31T^{2} \)
37 \( 1 + 8.16T + 37T^{2} \)
41 \( 1 - 5.59T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 9.28T + 53T^{2} \)
59 \( 1 - 6.89T + 59T^{2} \)
61 \( 1 - 8.50T + 61T^{2} \)
67 \( 1 - 7.61T + 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 + 4.83T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 + 6.56T + 97T^{2} \)
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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

−7.27210193707051229551659551084, −6.75951130383761505048047065203, −5.85430533882383361643687803157, −5.35758966007766713949006977439, −4.60834196621281012460405749998, −3.94054620519324077956362912130, −3.51972012127801356651264369315, −2.41221243051754464504885238777, −1.81510793660135053301615531770, 0, 1.81510793660135053301615531770, 2.41221243051754464504885238777, 3.51972012127801356651264369315, 3.94054620519324077956362912130, 4.60834196621281012460405749998, 5.35758966007766713949006977439, 5.85430533882383361643687803157, 6.75951130383761505048047065203, 7.27210193707051229551659551084

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