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Properties

Label 2-3024-63.4-c1-0-30
Degree $2$
Conductor $3024$
Sign $0.983 + 0.181i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.69·5-s + (2.60 + 0.436i)7-s − 0.892·11-s + (0.598 − 1.03i)13-s + (0.124 − 0.216i)17-s + (−1.40 − 2.43i)19-s + 2.47·23-s + 8.63·25-s + (−2.07 − 3.58i)29-s + (1.79 + 3.10i)31-s + (9.63 + 1.61i)35-s + (−2.36 − 4.09i)37-s + (2.39 − 4.14i)41-s + (4.98 + 8.64i)43-s + (5.08 − 8.81i)47-s + ⋯
L(s)  = 1  + 1.65·5-s + (0.986 + 0.165i)7-s − 0.269·11-s + (0.165 − 0.287i)13-s + (0.0303 − 0.0525i)17-s + (−0.322 − 0.557i)19-s + 0.516·23-s + 1.72·25-s + (−0.384 − 0.666i)29-s + (0.321 + 0.557i)31-s + (1.62 + 0.272i)35-s + (−0.388 − 0.673i)37-s + (0.373 − 0.646i)41-s + (0.760 + 1.31i)43-s + (0.741 − 1.28i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.968959756\)
\(L(\frac12)\) \(\approx\) \(2.968959756\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.436i)T \)
good5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 + 0.892T + 11T^{2} \)
13 \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.124 + 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 - 3.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.98 - 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.94 + 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.40 - 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.514 - 0.891i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (0.915 - 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.899 - 1.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.20 - 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.877309826473988690340244880104, −8.024100730212612000693276646763, −7.14527910434215888380750638831, −6.32785497126316397794422330964, −5.49907070260286821480693956112, −5.14213859224445401171137069671, −4.09462624731068945802589664849, −2.69798254483264302472268735913, −2.10343093901933944688236311368, −1.07309205254490532184112002194, 1.24700007580303937119236309587, 1.97440507665854235878317149291, 2.86962607826944326909164085834, 4.19271619975966582691733077588, 5.01059007979840123319937469866, 5.71492155739912026520233577654, 6.30300765909995686948182453670, 7.25110296618037797253610378989, 8.005295580089303755052072553893, 8.959065736527253661709924883723

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