Properties

Degree $2$
Conductor $3024$
Sign $-0.783 - 0.620i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.481·5-s + (−2.53 − 0.763i)7-s + 3.38·11-s + (−2.86 − 4.95i)13-s + (−2.75 − 4.77i)17-s + (−2.18 + 3.77i)19-s + 3.62·23-s − 4.76·25-s + (−1.53 + 2.65i)29-s + (−4.67 + 8.09i)31-s + (−1.21 − 0.367i)35-s + (1.48 − 2.57i)37-s + (6.29 + 10.9i)41-s + (−1.90 + 3.30i)43-s + (1.88 + 3.26i)47-s + ⋯
L(s)  = 1  + 0.215·5-s + (−0.957 − 0.288i)7-s + 1.01·11-s + (−0.793 − 1.37i)13-s + (−0.668 − 1.15i)17-s + (−0.500 + 0.866i)19-s + 0.756·23-s − 0.953·25-s + (−0.284 + 0.492i)29-s + (−0.839 + 1.45i)31-s + (−0.206 − 0.0621i)35-s + (0.244 − 0.422i)37-s + (0.983 + 1.70i)41-s + (−0.291 + 0.504i)43-s + (0.274 + 0.475i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.783 - 0.620i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.783 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2490364358\)
\(L(\frac12)\) \(\approx\) \(0.2490364358\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 0.763i)T \)
good5 \( 1 - 0.481T + 5T^{2} \)
11 \( 1 - 3.38T + 11T^{2} \)
13 \( 1 + (2.86 + 4.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.75 + 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.18 - 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + (1.53 - 2.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.67 - 8.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.48 + 2.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.29 - 10.9i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 - 3.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.88 - 3.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.57 + 9.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.21 - 7.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.64 - 6.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.28 + 2.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.94T + 71T^{2} \)
73 \( 1 + (0.862 + 1.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.79 + 4.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.119 - 0.206i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.648 - 1.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.02 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.331657166425241926090472895316, −8.250706200778242315532480193801, −7.39624571050342819777446089821, −6.79818274900007493252529092898, −6.03486657500621236986687067989, −5.24388426647061864850149798009, −4.30157318345812968014129521894, −3.34653307011039098456212636240, −2.66454401131630348479397670340, −1.27374328051858899578293342001, 0.07702337051673516838435012330, 1.82564729899444136295054638111, 2.52023273341085959956108773289, 3.91445365652312425446431771591, 4.22191636996191113239162056934, 5.49521804700775310026405170367, 6.32790314142629148050078311232, 6.74807066003691686294918886238, 7.53292664867747033761832278657, 8.692934904149241504386384689010

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