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 + 2.93·5-s + 1.35·7-s + 9-s − 0.0643·11-s − 2.07·13-s + 2.93·15-s − 1.73·17-s + 7.35·19-s + 1.35·21-s − 0.641·23-s + 3.60·25-s + 27-s + 3.65·29-s − 3.92·31-s − 0.0643·33-s + 3.97·35-s + 8.28·37-s − 2.07·39-s + 1.55·41-s − 2.77·43-s + 2.93·45-s − 4.51·47-s − 5.15·49-s − 1.73·51-s − 1.55·53-s − 0.188·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.31·5-s + 0.512·7-s + 0.333·9-s − 0.0193·11-s − 0.574·13-s + 0.757·15-s − 0.421·17-s + 1.68·19-s + 0.296·21-s − 0.133·23-s + 0.720·25-s + 0.192·27-s + 0.678·29-s − 0.704·31-s − 0.0111·33-s + 0.672·35-s + 1.36·37-s − 0.331·39-s + 0.242·41-s − 0.423·43-s + 0.437·45-s − 0.658·47-s − 0.737·49-s − 0.243·51-s − 0.212·53-s − 0.0254·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.439160182$
$L(\frac12)$  $\approx$  $3.439160182$
$L(\frac{3}{2})$   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 \( 1 - 2.93T + 5T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 + 0.0643T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 7.35T + 19T^{2} \)
23 \( 1 + 0.641T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 3.92T + 31T^{2} \)
37 \( 1 - 8.28T + 37T^{2} \)
41 \( 1 - 1.55T + 41T^{2} \)
43 \( 1 + 2.77T + 43T^{2} \)
47 \( 1 + 4.51T + 47T^{2} \)
53 \( 1 + 1.55T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 4.50T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 6.16T + 73T^{2} \)
79 \( 1 - 2.51T + 79T^{2} \)
83 \( 1 - 1.74T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−8.463202031289279606313746472327, −7.76871283307134278589756627926, −7.02224178749497908290951995564, −6.24549598306575649522017577461, −5.35080407644551659479628507937, −4.88547522941156559694750400927, −3.75913834829276337498812270594, −2.73306865778589576369929732117, −2.07753994388778658422917351544, −1.11343893244387602947438037759, 1.11343893244387602947438037759, 2.07753994388778658422917351544, 2.73306865778589576369929732117, 3.75913834829276337498812270594, 4.88547522941156559694750400927, 5.35080407644551659479628507937, 6.24549598306575649522017577461, 7.02224178749497908290951995564, 7.76871283307134278589756627926, 8.463202031289279606313746472327

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