Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 0.716·5-s − 3.33·7-s + 9-s − 1.47·11-s − 1.07·13-s − 0.716·15-s + 4.68·17-s − 1.60·19-s − 3.33·21-s + 5.96·23-s − 4.48·25-s + 27-s + 4.73·29-s + 8.82·31-s − 1.47·33-s + 2.38·35-s − 10.4·37-s − 1.07·39-s − 2.92·41-s − 8.38·43-s − 0.716·45-s + 6.24·47-s + 4.08·49-s + 4.68·51-s − 9.50·53-s + 1.05·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.320·5-s − 1.25·7-s + 0.333·9-s − 0.443·11-s − 0.297·13-s − 0.184·15-s + 1.13·17-s − 0.367·19-s − 0.726·21-s + 1.24·23-s − 0.897·25-s + 0.192·27-s + 0.878·29-s + 1.58·31-s − 0.256·33-s + 0.403·35-s − 1.71·37-s − 0.171·39-s − 0.456·41-s − 1.27·43-s − 0.106·45-s + 0.910·47-s + 0.584·49-s + 0.656·51-s − 1.30·53-s + 0.142·55-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  =  0
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.682102820$
$L(\frac12)$  $\approx$  $1.682102820$
$L(\frac{3}{2})$   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 \( 1 + 0.716T + 5T^{2} \)
7 \( 1 + 3.33T + 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
23 \( 1 - 5.96T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 8.82T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 2.92T + 41T^{2} \)
43 \( 1 + 8.38T + 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
59 \( 1 + 4.74T + 59T^{2} \)
61 \( 1 - 7.09T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 - 7.30T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 1.24T + 83T^{2} \)
89 \( 1 + 7.67T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.337876787299501561504173275636, −7.86743269956584880668784063275, −6.88375913224511693908078860284, −6.50475893188823352760808405089, −5.40399836750024361719032491916, −4.67015891328335782807944683512, −3.46472421167404784525848669852, −3.20823190650998150418392019065, −2.15571078608387514564322932728, −0.70454365069032092724161603499, 0.70454365069032092724161603499, 2.15571078608387514564322932728, 3.20823190650998150418392019065, 3.46472421167404784525848669852, 4.67015891328335782807944683512, 5.40399836750024361719032491916, 6.50475893188823352760808405089, 6.88375913224511693908078860284, 7.86743269956584880668784063275, 8.337876787299501561504173275636

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