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 − 3.64·5-s − 1.45·7-s + 9-s − 6.42·11-s − 4.46·13-s − 3.64·15-s − 3.08·17-s − 1.23·19-s − 1.45·21-s + 1.06·23-s + 8.24·25-s + 27-s + 1.27·29-s + 4.58·31-s − 6.42·33-s + 5.28·35-s + 3.59·37-s − 4.46·39-s − 9.41·41-s + 1.44·43-s − 3.64·45-s − 3.82·47-s − 4.89·49-s − 3.08·51-s + 9.67·53-s + 23.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.62·5-s − 0.548·7-s + 0.333·9-s − 1.93·11-s − 1.23·13-s − 0.939·15-s − 0.748·17-s − 0.282·19-s − 0.316·21-s + 0.222·23-s + 1.64·25-s + 0.192·27-s + 0.236·29-s + 0.823·31-s − 1.11·33-s + 0.893·35-s + 0.591·37-s − 0.715·39-s − 1.46·41-s + 0.220·43-s − 0.542·45-s − 0.557·47-s − 0.698·49-s − 0.432·51-s + 1.32·53-s + 3.15·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5861424980$
$L(\frac12)$  $\approx$  $0.5861424980$
$L(\frac{3}{2})$   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 \( 1 + 3.64T + 5T^{2} \)
7 \( 1 + 1.45T + 7T^{2} \)
11 \( 1 + 6.42T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 + 3.08T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 1.06T + 23T^{2} \)
29 \( 1 - 1.27T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 - 1.44T + 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 - 9.67T + 53T^{2} \)
59 \( 1 - 7.68T + 59T^{2} \)
61 \( 1 - 6.50T + 61T^{2} \)
67 \( 1 + 7.73T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 2.25T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 - 3.71T + 89T^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−8.281796732235196041658357359032, −7.76660262393015367882605360588, −7.24052667504083940302868414894, −6.50242476221095606715892489477, −5.13630945406012565435634268285, −4.67844470071681583362286748744, −3.76803132922314500768506889386, −2.91608994315836555009535425993, −2.36345706960652270200607565194, −0.39380280619834365659702284329, 0.39380280619834365659702284329, 2.36345706960652270200607565194, 2.91608994315836555009535425993, 3.76803132922314500768506889386, 4.67844470071681583362286748744, 5.13630945406012565435634268285, 6.50242476221095606715892489477, 7.24052667504083940302868414894, 7.76660262393015367882605360588, 8.281796732235196041658357359032

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