Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.377 + 0.925i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44i·5-s + (−1 − 2.44i)7-s + 2.44i·11-s + 4.89i·13-s + 2.44i·17-s + 2·19-s − 7.34i·23-s − 0.999·25-s + 6·29-s + 8·31-s + (−5.99 + 2.44i)35-s − 4·37-s − 7.34i·41-s + 4.89i·43-s + 12·47-s + ⋯
L(s)  = 1  − 1.09i·5-s + (−0.377 − 0.925i)7-s + 0.738i·11-s + 1.35i·13-s + 0.594i·17-s + 0.458·19-s − 1.53i·23-s − 0.199·25-s + 1.11·29-s + 1.43·31-s + (−1.01 + 0.414i)35-s − 0.657·37-s − 1.14i·41-s + 0.747i·43-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.377 + 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.377 + 0.925i)$
$L(1)$  $\approx$  $1.769934436$
$L(\frac12)$  $\approx$  $1.769934436$
$L(\frac{3}{2})$   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 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + 2.44iT - 5T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 - 2.44iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 7.34iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 7.34iT - 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 + 9.79iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 12.2iT - 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−8.503764164737708725993132932199, −7.53688459541587718216138216824, −6.79004559055561789030135910470, −6.28730855364845039368042757494, −5.08551732690939680305511450819, −4.40567675297270199358108154002, −4.05633313539108606858140977856, −2.71239194733128106647376662670, −1.59666040841639956781098124158, −0.65312076414784849846568338763, 0.947852183210974502938628026102, 2.59502862412262800116140544329, 2.95363662569522248918609961315, 3.68441074831981262996201358204, 5.10885848877651099318950978038, 5.63001794155721557041355695650, 6.37479369745579898372395713839, 7.04553103045493731933468230575, 7.900989008141513844519779175484, 8.469204947544295420021501899176

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