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 − 2.99·5-s − 4.65·7-s + 9-s + 0.962·11-s + 0.976·13-s − 2.99·15-s + 2.56·17-s + 8.42·19-s − 4.65·21-s + 6.29·23-s + 3.96·25-s + 27-s − 3.04·29-s − 10.1·31-s + 0.962·33-s + 13.9·35-s + 10.7·37-s + 0.976·39-s + 0.413·41-s − 5.35·43-s − 2.99·45-s − 8.47·47-s + 14.7·49-s + 2.56·51-s − 13.1·53-s − 2.88·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.33·5-s − 1.76·7-s + 0.333·9-s + 0.290·11-s + 0.270·13-s − 0.773·15-s + 0.621·17-s + 1.93·19-s − 1.01·21-s + 1.31·23-s + 0.792·25-s + 0.192·27-s − 0.565·29-s − 1.82·31-s + 0.167·33-s + 2.35·35-s + 1.76·37-s + 0.156·39-s + 0.0645·41-s − 0.816·43-s − 0.446·45-s − 1.23·47-s + 2.10·49-s + 0.358·51-s − 1.80·53-s − 0.388·55-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 \( 1 + 2.99T + 5T^{2} \)
7 \( 1 + 4.65T + 7T^{2} \)
11 \( 1 - 0.962T + 11T^{2} \)
13 \( 1 - 0.976T + 13T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 - 8.42T + 19T^{2} \)
23 \( 1 - 6.29T + 23T^{2} \)
29 \( 1 + 3.04T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 0.413T + 41T^{2} \)
43 \( 1 + 5.35T + 43T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 0.101T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 1.47T + 79T^{2} \)
83 \( 1 + 4.56T + 83T^{2} \)
89 \( 1 - 8.20T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−7.78404155726340048331700017292, −7.55583005159352344158407145798, −6.79248659241408916929437875200, −5.97786728346179991070893941713, −4.98741262851435444414738527203, −3.92887789007512121599471926072, −3.19355338488292817630320468877, −3.11930735800683632948912094801, −1.25977988748857732524129212862, 0, 1.25977988748857732524129212862, 3.11930735800683632948912094801, 3.19355338488292817630320468877, 3.92887789007512121599471926072, 4.98741262851435444414738527203, 5.97786728346179991070893941713, 6.79248659241408916929437875200, 7.55583005159352344158407145798, 7.78404155726340048331700017292

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