Properties

Degree $2$
Conductor $3024$
Sign $0.794 - 0.607i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.58·5-s + (1.80 + 1.93i)7-s + 5.17·11-s + (−0.681 − 1.18i)13-s + (2.30 + 3.99i)17-s + (−0.0321 + 0.0557i)19-s + 6.74·23-s − 2.49·25-s + (−4.70 + 8.15i)29-s + (−1.33 + 2.30i)31-s + (2.86 + 3.05i)35-s + (0.880 − 1.52i)37-s + (0.858 + 1.48i)41-s + (5.12 − 8.86i)43-s + (−2.60 − 4.51i)47-s + ⋯
L(s)  = 1  + 0.707·5-s + (0.683 + 0.729i)7-s + 1.55·11-s + (−0.189 − 0.327i)13-s + (0.559 + 0.969i)17-s + (−0.00738 + 0.0127i)19-s + 1.40·23-s − 0.499·25-s + (−0.874 + 1.51i)29-s + (−0.239 + 0.414i)31-s + (0.483 + 0.516i)35-s + (0.144 − 0.250i)37-s + (0.134 + 0.232i)41-s + (0.780 − 1.35i)43-s + (−0.379 − 0.657i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.794 - 0.607i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.632867532\)
\(L(\frac12)\) \(\approx\) \(2.632867532\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 1.93i)T \)
good5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + (0.681 + 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.30 - 3.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0321 - 0.0557i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 + (4.70 - 8.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.33 - 2.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.858 - 1.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.12 + 8.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.60 + 4.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.479 - 0.831i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.66 + 8.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.19 + 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.24 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (0.941 + 1.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.26 - 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 - 8.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.12 + 7.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.26 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.008858341231727974916655296529, −8.178383820736902133246053581553, −7.22295058252846456836800553911, −6.50152116220563159271351565746, −5.63428378349700900043019988594, −5.18736872073735150175693109342, −4.04584332298345808672355732608, −3.19860806882596439456816359526, −1.95444687710554939861871887616, −1.31141764168966272803955484920, 0.946365782672896600178612914159, 1.78194094722549470402193452124, 2.95399925244838482683138147136, 4.08987778032991241787439452160, 4.62496103427935233841868377457, 5.68222174368690564668128742064, 6.33617988929331468960107284716, 7.26742335010648946485821735926, 7.67370627383732243665424734348, 8.846117958492853530240788601561

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