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 − 2.09·5-s + 2.97·7-s + 9-s − 0.138·11-s + 5.06·13-s − 2.09·15-s + 0.299·17-s + 0.432·19-s + 2.97·21-s + 5.72·23-s − 0.590·25-s + 27-s − 0.718·29-s − 4.27·31-s − 0.138·33-s − 6.24·35-s − 3.92·37-s + 5.06·39-s + 7.57·41-s + 12.3·43-s − 2.09·45-s − 9.22·47-s + 1.84·49-s + 0.299·51-s + 1.26·53-s + 0.290·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.939·5-s + 1.12·7-s + 0.333·9-s − 0.0416·11-s + 1.40·13-s − 0.542·15-s + 0.0725·17-s + 0.0991·19-s + 0.649·21-s + 1.19·23-s − 0.118·25-s + 0.192·27-s − 0.133·29-s − 0.767·31-s − 0.0240·33-s − 1.05·35-s − 0.644·37-s + 0.810·39-s + 1.18·41-s + 1.88·43-s − 0.313·45-s − 1.34·47-s + 0.263·49-s + 0.0418·51-s + 0.173·53-s + 0.0391·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.536896854$
$L(\frac12)$  $\approx$  $2.536896854$
$L(\frac{3}{2})$   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 \( 1 + 2.09T + 5T^{2} \)
7 \( 1 - 2.97T + 7T^{2} \)
11 \( 1 + 0.138T + 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 - 0.299T + 17T^{2} \)
19 \( 1 - 0.432T + 19T^{2} \)
23 \( 1 - 5.72T + 23T^{2} \)
29 \( 1 + 0.718T + 29T^{2} \)
31 \( 1 + 4.27T + 31T^{2} \)
37 \( 1 + 3.92T + 37T^{2} \)
41 \( 1 - 7.57T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + 9.22T + 47T^{2} \)
53 \( 1 - 1.26T + 53T^{2} \)
59 \( 1 - 3.01T + 59T^{2} \)
61 \( 1 - 2.98T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 7.07T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−8.552090552281594003140546002294, −7.65161149761981983086604098525, −7.38186744423141795788983161339, −6.27167167945800016993093429141, −5.37115083932913911178380433842, −4.50035479978729527485243106026, −3.84368491960950214068128333931, −3.11411094440532902444973489000, −1.90968660246799491948094041128, −0.944889107847477811213670416716, 0.944889107847477811213670416716, 1.90968660246799491948094041128, 3.11411094440532902444973489000, 3.84368491960950214068128333931, 4.50035479978729527485243106026, 5.37115083932913911178380433842, 6.27167167945800016993093429141, 7.38186744423141795788983161339, 7.65161149761981983086604098525, 8.552090552281594003140546002294

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