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  − 2.23·2-s + 3.00·4-s + 2·5-s − 7-s − 2.23·8-s − 4.47·10-s + 1.23·13-s + 2.23·14-s − 0.999·16-s + 1.23·17-s + 2.47·19-s + 6.00·20-s + 6.47·23-s − 25-s − 2.76·26-s − 3.00·28-s − 0.472·29-s − 7.23·31-s + 6.70·32-s − 2.76·34-s − 2·35-s + 0.472·37-s − 5.52·38-s − 4.47·40-s − 6.76·41-s − 8·43-s − 14.4·46-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s + 0.894·5-s − 0.377·7-s − 0.790·8-s − 1.41·10-s + 0.342·13-s + 0.597·14-s − 0.249·16-s + 0.299·17-s + 0.567·19-s + 1.34·20-s + 1.34·23-s − 0.200·25-s − 0.542·26-s − 0.566·28-s − 0.0876·29-s − 1.29·31-s + 1.18·32-s − 0.474·34-s − 0.338·35-s + 0.0776·37-s − 0.896·38-s − 0.707·40-s − 1.05·41-s − 1.21·43-s − 2.13·46-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 \( 1 + 2.23T + 2T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + 0.472T + 29T^{2} \)
31 \( 1 + 7.23T + 31T^{2} \)
37 \( 1 - 0.472T + 37T^{2} \)
41 \( 1 + 6.76T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 3.23T + 59T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 - 5.23T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−7.72862752904650541078578623544, −6.84178033878918086473019447893, −6.54585758853931192056088207643, −5.55162246720538219839015970625, −4.95672095849414624037505487683, −3.63534980479515226249206214833, −2.84620157227462634890446400260, −1.83697598738312793209164987775, −1.24665757722581832356087043882, 0, 1.24665757722581832356087043882, 1.83697598738312793209164987775, 2.84620157227462634890446400260, 3.63534980479515226249206214833, 4.95672095849414624037505487683, 5.55162246720538219839015970625, 6.54585758853931192056088207643, 6.84178033878918086473019447893, 7.72862752904650541078578623544

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