Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 167 $
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 + 0.413·5-s + 2.72·7-s + 9-s − 3.29·11-s − 3.51·13-s + 0.413·15-s − 2.04·17-s + 0.104·19-s + 2.72·21-s − 7.14·23-s − 4.82·25-s + 27-s + 3.94·29-s − 6.72·31-s − 3.29·33-s + 1.12·35-s − 11.8·37-s − 3.51·39-s + 1.68·41-s − 3.25·43-s + 0.413·45-s − 1.28·47-s + 0.427·49-s − 2.04·51-s − 2.39·53-s − 1.36·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.185·5-s + 1.03·7-s + 0.333·9-s − 0.992·11-s − 0.974·13-s + 0.106·15-s − 0.495·17-s + 0.0239·19-s + 0.594·21-s − 1.49·23-s − 0.965·25-s + 0.192·27-s + 0.733·29-s − 1.20·31-s − 0.573·33-s + 0.190·35-s − 1.94·37-s − 0.562·39-s + 0.263·41-s − 0.495·43-s + 0.0616·45-s − 0.187·47-s + 0.0610·49-s − 0.286·51-s − 0.328·53-s − 0.183·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 - 0.413T + 5T^{2} \)
7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 + 3.29T + 11T^{2} \)
13 \( 1 + 3.51T + 13T^{2} \)
17 \( 1 + 2.04T + 17T^{2} \)
19 \( 1 - 0.104T + 19T^{2} \)
23 \( 1 + 7.14T + 23T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 + 6.72T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 + 3.25T + 43T^{2} \)
47 \( 1 + 1.28T + 47T^{2} \)
53 \( 1 + 2.39T + 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 + 0.869T + 61T^{2} \)
67 \( 1 - 0.884T + 67T^{2} \)
71 \( 1 - 5.76T + 71T^{2} \)
73 \( 1 + 9.55T + 73T^{2} \)
79 \( 1 - 8.30T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−8.047712377696303594423930534243, −7.56727177453783531374626081508, −6.78507900216073861859975559366, −5.68502139194502149534383658410, −5.04918407102090649505057626287, −4.33331891110010956105727980367, −3.36650408746486904357079300050, −2.24303317049076766486315318735, −1.81871984208376143198129717124, 0, 1.81871984208376143198129717124, 2.24303317049076766486315318735, 3.36650408746486904357079300050, 4.33331891110010956105727980367, 5.04918407102090649505057626287, 5.68502139194502149534383658410, 6.78507900216073861859975559366, 7.56727177453783531374626081508, 8.047712377696303594423930534243

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