Properties

Degree 2
Conductor $ 3 \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  + 0.618·2-s − 3-s − 1.61·4-s + 2.23·5-s − 0.618·6-s − 7-s − 2.23·8-s + 9-s + 1.38·10-s + 1.61·12-s + 4.23·13-s − 0.618·14-s − 2.23·15-s + 1.85·16-s − 2·17-s + 0.618·18-s − 7.47·19-s − 3.61·20-s + 21-s − 2·23-s + 2.23·24-s + 2.61·26-s − 27-s + 1.61·28-s − 5·29-s − 1.38·30-s + 1.52·31-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.999·5-s − 0.252·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.437·10-s + 0.467·12-s + 1.17·13-s − 0.165·14-s − 0.577·15-s + 0.463·16-s − 0.485·17-s + 0.145·18-s − 1.71·19-s − 0.809·20-s + 0.218·21-s − 0.417·23-s + 0.456·24-s + 0.513·26-s − 0.192·27-s + 0.305·28-s − 0.928·29-s − 0.252·30-s + 0.274·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2541} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 0.618T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
13 \( 1 - 4.23T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 7.47T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 - 7.47T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 - 0.527T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 + 9.94T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 8.47T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−8.724420240516919386783489014798, −7.82459885775674357039920418760, −6.50464694666663581733534558096, −6.06938857426823510828172508690, −5.59365314835826669698926045518, −4.43604560350728737821489722897, −3.98360891822452192730159137124, −2.71431353088794845757354849048, −1.50731509326314313504178088799, 0, 1.50731509326314313504178088799, 2.71431353088794845757354849048, 3.98360891822452192730159137124, 4.43604560350728737821489722897, 5.59365314835826669698926045518, 6.06938857426823510828172508690, 6.50464694666663581733534558096, 7.82459885775674357039920418760, 8.724420240516919386783489014798

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