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 − 0.858·5-s + 2.33·7-s + 9-s + 0.664·11-s − 2.80·13-s − 0.858·15-s − 4.73·17-s − 7.58·19-s + 2.33·21-s + 6.58·23-s − 4.26·25-s + 27-s − 4.75·29-s − 2.96·31-s + 0.664·33-s − 2.00·35-s + 4.12·37-s − 2.80·39-s − 7.76·41-s + 4.13·43-s − 0.858·45-s − 1.39·47-s − 1.56·49-s − 4.73·51-s − 1.74·53-s − 0.569·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.383·5-s + 0.881·7-s + 0.333·9-s + 0.200·11-s − 0.777·13-s − 0.221·15-s − 1.14·17-s − 1.74·19-s + 0.508·21-s + 1.37·23-s − 0.852·25-s + 0.192·27-s − 0.883·29-s − 0.531·31-s + 0.115·33-s − 0.338·35-s + 0.678·37-s − 0.449·39-s − 1.21·41-s + 0.631·43-s − 0.127·45-s − 0.202·47-s − 0.222·49-s − 0.663·51-s − 0.239·53-s − 0.0768·55-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 \( 1 + 0.858T + 5T^{2} \)
7 \( 1 - 2.33T + 7T^{2} \)
11 \( 1 - 0.664T + 11T^{2} \)
13 \( 1 + 2.80T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + 7.58T + 19T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 + 7.76T + 41T^{2} \)
43 \( 1 - 4.13T + 43T^{2} \)
47 \( 1 + 1.39T + 47T^{2} \)
53 \( 1 + 1.74T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 0.762T + 61T^{2} \)
67 \( 1 - 0.00130T + 67T^{2} \)
71 \( 1 + 2.99T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 + 4.94T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.082366029677731649925088905707, −7.45598752554869832252201879124, −6.79663946881233899451856346124, −5.89574581005216367969722322807, −4.71599602193156833328120926482, −4.44322900350144354987282229121, −3.42954198585822308054046392617, −2.35119402809370291308673280722, −1.68017941238334238094147569791, 0, 1.68017941238334238094147569791, 2.35119402809370291308673280722, 3.42954198585822308054046392617, 4.44322900350144354987282229121, 4.71599602193156833328120926482, 5.89574581005216367969722322807, 6.79663946881233899451856346124, 7.45598752554869832252201879124, 8.082366029677731649925088905707

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