Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
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-s − 5-s − 3·7-s + 9-s − 2·11-s − 2·13-s − 15-s − 4·17-s − 4·19-s − 3·21-s + 7·23-s − 4·25-s + 27-s − 8·29-s + 3·31-s − 2·33-s + 3·35-s − 3·37-s − 2·39-s + 41-s − 11·43-s − 45-s + 2·49-s − 4·51-s + 11·53-s + 2·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s − 0.654·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 0.348·33-s + 0.507·35-s − 0.493·37-s − 0.320·39-s + 0.156·41-s − 1.67·43-s − 0.149·45-s + 2/7·49-s − 0.560·51-s + 1.51·53-s + 0.269·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{804} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
67 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.750234589912995839555802068068, −9.010124849067987371032486144526, −8.199712549586293443059014780518, −7.19634459883027179171790840316, −6.58469246152879980363958305254, −5.31649357921964130503329570462, −4.18065040315259304688251616469, −3.23320972721445320648750162796, −2.22330826788334876611733342610, 0, 2.22330826788334876611733342610, 3.23320972721445320648750162796, 4.18065040315259304688251616469, 5.31649357921964130503329570462, 6.58469246152879980363958305254, 7.19634459883027179171790840316, 8.199712549586293443059014780518, 9.010124849067987371032486144526, 9.750234589912995839555802068068

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