Properties

Degree $2$
Conductor $3024$
Sign $-0.962 - 0.272i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.84·5-s + (−0.676 − 2.55i)7-s + 1.80·11-s + (−0.692 − 1.19i)13-s + (0.833 + 1.44i)17-s + (0.0802 − 0.138i)19-s + 3.20·23-s + 9.75·25-s + (3.78 − 6.54i)29-s + (1.61 − 2.78i)31-s + (2.59 + 9.82i)35-s + (1.58 − 2.74i)37-s + (−6.00 − 10.3i)41-s + (−3.45 + 5.98i)43-s + (−5.71 − 9.90i)47-s + ⋯
L(s)  = 1  − 1.71·5-s + (−0.255 − 0.966i)7-s + 0.544·11-s + (−0.192 − 0.332i)13-s + (0.202 + 0.350i)17-s + (0.0184 − 0.0318i)19-s + 0.667·23-s + 1.95·25-s + (0.701 − 1.21i)29-s + (0.289 − 0.500i)31-s + (0.439 + 1.66i)35-s + (0.260 − 0.451i)37-s + (−0.937 − 1.62i)41-s + (−0.526 + 0.912i)43-s + (−0.834 − 1.44i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.962 - 0.272i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.962 - 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2248875395\)
\(L(\frac12)\) \(\approx\) \(0.2248875395\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.676 + 2.55i)T \)
good5 \( 1 + 3.84T + 5T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.833 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0802 + 0.138i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.20T + 23T^{2} \)
29 \( 1 + (-3.78 + 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.61 + 2.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.00 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.71 + 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.37 + 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.53 - 13.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.60 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.03 - 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.45 - 2.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.04 - 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.235621590077620660530435504688, −7.49933736022890517245023261950, −7.06170996804499334038759846580, −6.21231871356699321529153292930, −4.99606936497273201620825131143, −4.14108055357223956268553136984, −3.75174080850847650479172746528, −2.81493812222992992855935402032, −1.10460887081312213889495058921, −0.085738505804238146699625907740, 1.43208103844999167433360697519, 3.03389930262620908854798088556, 3.35930860549660982925328506158, 4.62468808182130379788377683126, 4.97577048002550194448756511538, 6.34449455301201861340103828054, 6.84752176500317445207668943801, 7.74108882734573881700296036165, 8.366428444876473106666425753278, 8.975914204640792568063474416025

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