Properties

Degree $2$
Conductor $3024$
Sign $-0.983 + 0.182i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.46 + 2.52i)5-s + (0.138 + 2.64i)7-s + (0.676 + 1.17i)11-s + (−0.733 − 1.26i)13-s + (−1.65 + 2.86i)17-s + (1.10 + 1.91i)19-s + (−1.31 + 2.27i)23-s + (−1.76 − 3.05i)25-s + (−0.521 + 0.903i)29-s − 3.27·31-s + (−6.88 − 3.50i)35-s + (5.43 + 9.41i)37-s + (0.904 + 1.56i)41-s + (2.17 − 3.76i)43-s + 3.97·47-s + ⋯
L(s)  = 1  + (−0.653 + 1.13i)5-s + (0.0523 + 0.998i)7-s + (0.204 + 0.353i)11-s + (−0.203 − 0.352i)13-s + (−0.401 + 0.695i)17-s + (0.253 + 0.438i)19-s + (−0.274 + 0.474i)23-s + (−0.353 − 0.611i)25-s + (−0.0968 + 0.167i)29-s − 0.588·31-s + (−1.16 − 0.592i)35-s + (0.893 + 1.54i)37-s + (0.141 + 0.244i)41-s + (0.331 − 0.573i)43-s + 0.580·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.983 + 0.182i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8676190017\)
\(L(\frac12)\) \(\approx\) \(0.8676190017\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.138 - 2.64i)T \)
good5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.676 - 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.733 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 - 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.521 - 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.904 - 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.17 + 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.97T + 47T^{2} \)
53 \( 1 + (-3.22 + 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 0.559T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.767T + 79T^{2} \)
83 \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.20 + 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.14 - 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.138686087507665353257822000811, −8.206135977876501320627949432956, −7.68972785122902003344268913294, −6.85047287168995714825296269301, −6.16773599670113623243333521336, −5.39877956151991132135646429288, −4.35573272472230346346647930115, −3.44474738411413597767095440384, −2.73608402703109717600855297468, −1.71604167520707212933826040187, 0.29585011907784838677023229108, 1.16539585155989519254863255410, 2.55917256881559712599871324618, 3.84628798841530984798338874738, 4.33769871585361277090750873407, 5.03685991287246706641762645556, 6.01844820475500451453419230993, 7.02347675577853560468040638851, 7.59481211467894513592732322307, 8.283932416741768104817555319649

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