Properties

Degree 2
Conductor $ 2^{3} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s − 2·7-s + 6·9-s − 11-s + 9·15-s − 6·17-s + 4·19-s + 6·21-s + 23-s + 4·25-s − 9·27-s − 8·29-s − 7·31-s + 3·33-s + 6·35-s − 37-s + 4·41-s + 6·43-s − 18·45-s − 8·47-s − 3·49-s + 18·51-s + 2·53-s + 3·55-s − 12·57-s − 59-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 0.301·11-s + 2.32·15-s − 1.45·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s + 4/5·25-s − 1.73·27-s − 1.48·29-s − 1.25·31-s + 0.522·33-s + 1.01·35-s − 0.164·37-s + 0.624·41-s + 0.914·43-s − 2.68·45-s − 1.16·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 0.404·55-s − 1.58·57-s − 0.130·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{88} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 88,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.39814407971896, −18.44840376758474, −17.63042282918627, −16.41501109154892, −16.02085093624741, −15.13896432548456, −13.16442893918529, −12.38704213422327, −11.36466946676257, −10.89204680724752, −9.401670233685537, −7.573565195405382, −6.610175306374676, −5.267180620239121, −3.921312004127912, 0, 3.921312004127912, 5.267180620239121, 6.610175306374676, 7.573565195405382, 9.401670233685537, 10.89204680724752, 11.36466946676257, 12.38704213422327, 13.16442893918529, 15.13896432548456, 16.02085093624741, 16.41501109154892, 17.63042282918627, 18.44840376758474, 19.39814407971896

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