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 − 1.48·5-s − 1.03·7-s + 9-s + 5.64·11-s − 4.47·13-s − 1.48·15-s − 6.13·17-s + 5.87·19-s − 1.03·21-s − 1.22·23-s − 2.80·25-s + 27-s − 6.91·29-s + 3.70·31-s + 5.64·33-s + 1.54·35-s − 6.62·37-s − 4.47·39-s + 1.85·41-s − 2.29·43-s − 1.48·45-s + 3.59·47-s − 5.92·49-s − 6.13·51-s − 0.299·53-s − 8.36·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.662·5-s − 0.392·7-s + 0.333·9-s + 1.70·11-s − 1.24·13-s − 0.382·15-s − 1.48·17-s + 1.34·19-s − 0.226·21-s − 0.255·23-s − 0.560·25-s + 0.192·27-s − 1.28·29-s + 0.664·31-s + 0.981·33-s + 0.260·35-s − 1.08·37-s − 0.717·39-s + 0.290·41-s − 0.350·43-s − 0.220·45-s + 0.523·47-s − 0.845·49-s − 0.859·51-s − 0.0411·53-s − 1.12·55-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 \( 1 + 1.48T + 5T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 6.13T + 17T^{2} \)
19 \( 1 - 5.87T + 19T^{2} \)
23 \( 1 + 1.22T + 23T^{2} \)
29 \( 1 + 6.91T + 29T^{2} \)
31 \( 1 - 3.70T + 31T^{2} \)
37 \( 1 + 6.62T + 37T^{2} \)
41 \( 1 - 1.85T + 41T^{2} \)
43 \( 1 + 2.29T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 0.299T + 53T^{2} \)
59 \( 1 - 0.102T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + 5.27T + 67T^{2} \)
71 \( 1 - 2.00T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 + 3.92T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
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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

−8.119884888349185442392469794756, −7.15843039228053126184270824646, −6.97057949578169473694952149784, −5.95730780915483295830675783480, −4.88572829276751503110756923526, −4.07944146284679489219247240254, −3.54870421830755294067481489400, −2.53260671561345394896616841098, −1.51092403242314701006793699660, 0, 1.51092403242314701006793699660, 2.53260671561345394896616841098, 3.54870421830755294067481489400, 4.07944146284679489219247240254, 4.88572829276751503110756923526, 5.95730780915483295830675783480, 6.97057949578169473694952149784, 7.15843039228053126184270824646, 8.119884888349185442392469794756

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