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 + 1.53·5-s − 0.915·7-s + 9-s + 3.09·11-s + 1.39·13-s + 1.53·15-s − 1.50·17-s − 4.21·19-s − 0.915·21-s + 4.95·23-s − 2.64·25-s + 27-s + 1.76·29-s + 3.79·31-s + 3.09·33-s − 1.40·35-s + 4.15·37-s + 1.39·39-s + 8.32·41-s + 8.45·43-s + 1.53·45-s − 6.81·47-s − 6.16·49-s − 1.50·51-s + 1.43·53-s + 4.74·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.685·5-s − 0.346·7-s + 0.333·9-s + 0.933·11-s + 0.387·13-s + 0.395·15-s − 0.363·17-s − 0.966·19-s − 0.199·21-s + 1.03·23-s − 0.529·25-s + 0.192·27-s + 0.328·29-s + 0.682·31-s + 0.539·33-s − 0.237·35-s + 0.682·37-s + 0.223·39-s + 1.30·41-s + 1.28·43-s + 0.228·45-s − 0.994·47-s − 0.880·49-s − 0.210·51-s + 0.196·53-s + 0.640·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.894802593$
$L(\frac12)$  $\approx$  $2.894802593$
$L(\frac{3}{2})$   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 \( 1 - 1.53T + 5T^{2} \)
7 \( 1 + 0.915T + 7T^{2} \)
11 \( 1 - 3.09T + 11T^{2} \)
13 \( 1 - 1.39T + 13T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
23 \( 1 - 4.95T + 23T^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
31 \( 1 - 3.79T + 31T^{2} \)
37 \( 1 - 4.15T + 37T^{2} \)
41 \( 1 - 8.32T + 41T^{2} \)
43 \( 1 - 8.45T + 43T^{2} \)
47 \( 1 + 6.81T + 47T^{2} \)
53 \( 1 - 1.43T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 3.76T + 61T^{2} \)
67 \( 1 - 7.30T + 67T^{2} \)
71 \( 1 - 0.883T + 71T^{2} \)
73 \( 1 + 0.891T + 73T^{2} \)
79 \( 1 + 1.12T + 79T^{2} \)
83 \( 1 - 4.26T + 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−8.517476438757462625762327212289, −7.84583965366549109025467280728, −6.76133619734500476706478614373, −6.42706700071803908356086763439, −5.58922565703144326483323174534, −4.49538707917253446675915682392, −3.87078643208851997285045797535, −2.85521582411763115159362087928, −2.06685459868209504507168099010, −0.992704419129737678826076666887, 0.992704419129737678826076666887, 2.06685459868209504507168099010, 2.85521582411763115159362087928, 3.87078643208851997285045797535, 4.49538707917253446675915682392, 5.58922565703144326483323174534, 6.42706700071803908356086763439, 6.76133619734500476706478614373, 7.84583965366549109025467280728, 8.517476438757462625762327212289

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