Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.156 - 0.987i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61i·5-s + (−2.61 − 0.414i)7-s + 2i·11-s − 4.77i·13-s + 3.06i·17-s + 4.14·19-s − 7.65i·23-s − 1.82·25-s + 3.65·29-s + 3.06·31-s + (1.08 − 6.82i)35-s + 7.65·37-s + 9.55i·41-s + 3.65i·43-s − 7.39·47-s + ⋯
L(s)  = 1  + 1.16i·5-s + (−0.987 − 0.156i)7-s + 0.603i·11-s − 1.32i·13-s + 0.742i·17-s + 0.950·19-s − 1.59i·23-s − 0.365·25-s + 0.679·29-s + 0.549·31-s + (0.182 − 1.15i)35-s + 1.25·37-s + 1.49i·41-s + 0.557i·43-s − 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.156 - 0.987i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.156 - 0.987i)$
$L(1)$  $\approx$  $1.483274452$
$L(\frac12)$  $\approx$  $1.483274452$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.61 + 0.414i)T \)
good5 \( 1 - 2.61iT - 5T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 - 3.06iT - 17T^{2} \)
19 \( 1 - 4.14T + 19T^{2} \)
23 \( 1 + 7.65iT - 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 9.55iT - 41T^{2} \)
43 \( 1 - 3.65iT - 43T^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 8.47T + 59T^{2} \)
61 \( 1 - 2.61iT - 61T^{2} \)
67 \( 1 + 15.6iT - 67T^{2} \)
71 \( 1 - 8.82iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 2.16iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−8.377751376143033233088410236499, −7.907969652178949365004334796315, −6.95647822765587905518763864301, −6.51416754905775326968063878929, −5.87666648161798336109407970459, −4.82802777713938889896251815630, −3.88320671383187596002585908261, −2.90366040021137227855991874528, −2.67290882222862017130548317206, −0.959768828220903411829932930217, 0.53312484116502612299258074107, 1.57369232916943217919340366304, 2.82703216514415358211034722334, 3.66744423522454597249413832134, 4.52958469482472445361121480325, 5.32458902588352390530203266638, 5.96076679497236115840906829929, 6.85085150730252899905860291745, 7.49187320423776679644622386613, 8.479084061972747912279654769354

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