Properties

Degree 2
Conductor $ 3 \cdot 29 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯
L(s)  = 1  + 2-s − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{87} (86, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 87,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5770586370\)
\(L(\frac12)\)  \(\approx\)  \(0.5770586370\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;29\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 - T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 - T + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−14.39892195706409998885986607529, −13.18133367873374302632506132995, −12.37546425848442860075315832869, −11.74152542706301551240275566622, −10.16119846875737190734833337722, −9.215490822664834597853334075143, −7.06152498257571751488998986929, −6.04035672910010770144714336001, −4.91310283918857793479423733862, −3.53215773738360323132247593606, 3.53215773738360323132247593606, 4.91310283918857793479423733862, 6.04035672910010770144714336001, 7.06152498257571751488998986929, 9.215490822664834597853334075143, 10.16119846875737190734833337722, 11.74152542706301551240275566622, 12.37546425848442860075315832869, 13.18133367873374302632506132995, 14.39892195706409998885986607529

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