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 + 1.94·5-s − 3.19·7-s + 9-s − 0.785·11-s − 4.88·13-s + 1.94·15-s + 2.74·17-s + 1.67·19-s − 3.19·21-s − 6.23·23-s − 1.21·25-s + 27-s − 0.902·29-s + 5.13·31-s − 0.785·33-s − 6.21·35-s + 0.279·37-s − 4.88·39-s + 1.15·41-s + 4.50·43-s + 1.94·45-s − 8.29·47-s + 3.22·49-s + 2.74·51-s − 6.24·53-s − 1.52·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.869·5-s − 1.20·7-s + 0.333·9-s − 0.236·11-s − 1.35·13-s + 0.502·15-s + 0.666·17-s + 0.384·19-s − 0.697·21-s − 1.30·23-s − 0.243·25-s + 0.192·27-s − 0.167·29-s + 0.922·31-s − 0.136·33-s − 1.05·35-s + 0.0459·37-s − 0.782·39-s + 0.180·41-s + 0.687·43-s + 0.289·45-s − 1.21·47-s + 0.460·49-s + 0.384·51-s − 0.858·53-s − 0.205·55-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 \( 1 - 1.94T + 5T^{2} \)
7 \( 1 + 3.19T + 7T^{2} \)
11 \( 1 + 0.785T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 - 1.67T + 19T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + 0.902T + 29T^{2} \)
31 \( 1 - 5.13T + 31T^{2} \)
37 \( 1 - 0.279T + 37T^{2} \)
41 \( 1 - 1.15T + 41T^{2} \)
43 \( 1 - 4.50T + 43T^{2} \)
47 \( 1 + 8.29T + 47T^{2} \)
53 \( 1 + 6.24T + 53T^{2} \)
59 \( 1 + 9.18T + 59T^{2} \)
61 \( 1 - 0.717T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 - 2.50T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 4.93T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 1.53T + 97T^{2} \)
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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

−7.893802420937623497374371556909, −7.52941315893114196976669006045, −6.48751316548784785501715155206, −6.00517513961871479456594062710, −5.13258907211079374674547008891, −4.20548640993672804094863414734, −3.15411019598286334172903177789, −2.61532274271596415639689815077, −1.63316053981271216236284161298, 0, 1.63316053981271216236284161298, 2.61532274271596415639689815077, 3.15411019598286334172903177789, 4.20548640993672804094863414734, 5.13258907211079374674547008891, 6.00517513961871479456594062710, 6.48751316548784785501715155206, 7.52941315893114196976669006045, 7.893802420937623497374371556909

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