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 + 0.588·5-s − 1.23·7-s + 9-s − 5.01·11-s + 4.86·13-s + 0.588·15-s − 6.70·17-s − 1.72·19-s − 1.23·21-s + 3.90·23-s − 4.65·25-s + 27-s + 8.98·29-s − 5.36·31-s − 5.01·33-s − 0.726·35-s + 1.07·37-s + 4.86·39-s + 6.22·41-s − 9.74·43-s + 0.588·45-s + 3.85·47-s − 5.47·49-s − 6.70·51-s − 7.57·53-s − 2.95·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.263·5-s − 0.466·7-s + 0.333·9-s − 1.51·11-s + 1.34·13-s + 0.151·15-s − 1.62·17-s − 0.395·19-s − 0.269·21-s + 0.814·23-s − 0.930·25-s + 0.192·27-s + 1.66·29-s − 0.964·31-s − 0.873·33-s − 0.122·35-s + 0.175·37-s + 0.778·39-s + 0.972·41-s − 1.48·43-s + 0.0877·45-s + 0.562·47-s − 0.782·49-s − 0.939·51-s − 1.04·53-s − 0.398·55-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 \( 1 - 0.588T + 5T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 + 5.01T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 + 6.70T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 - 8.98T + 29T^{2} \)
31 \( 1 + 5.36T + 31T^{2} \)
37 \( 1 - 1.07T + 37T^{2} \)
41 \( 1 - 6.22T + 41T^{2} \)
43 \( 1 + 9.74T + 43T^{2} \)
47 \( 1 - 3.85T + 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 + 8.99T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + 6.37T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + 6.10T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 5.15T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.191312342435979008559525759622, −7.45078628114522855334829188939, −6.51565708580391851802761646101, −6.04445285854373882460791257392, −4.97986598903896824851853582442, −4.26573089777631933413270067927, −3.22924559491694813465683048314, −2.58945563122126296250162429783, −1.59523734853418015635971086764, 0, 1.59523734853418015635971086764, 2.58945563122126296250162429783, 3.22924559491694813465683048314, 4.26573089777631933413270067927, 4.97986598903896824851853582442, 6.04445285854373882460791257392, 6.51565708580391851802761646101, 7.45078628114522855334829188939, 8.191312342435979008559525759622

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