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Properties

Label 2-3024-9.4-c1-0-24
Degree $2$
Conductor $3024$
Sign $-0.199 + 0.979i$
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  + (−1.81 + 3.13i)5-s + (0.5 + 0.866i)7-s + (−1.95 − 3.39i)11-s + (−2.53 + 4.39i)13-s − 1.03·17-s + 2.50·19-s + (−2.47 + 4.29i)23-s + (−4.06 − 7.04i)25-s + (−4.60 − 7.97i)29-s + (−0.422 + 0.731i)31-s − 3.62·35-s + 4.84·37-s + (−2.07 + 3.59i)41-s + (−2.20 − 3.81i)43-s + (3.93 + 6.82i)47-s + ⋯
L(s)  = 1  + (−0.810 + 1.40i)5-s + (0.188 + 0.327i)7-s + (−0.590 − 1.02i)11-s + (−0.703 + 1.21i)13-s − 0.250·17-s + 0.575·19-s + (−0.516 + 0.895i)23-s + (−0.813 − 1.40i)25-s + (−0.854 − 1.48i)29-s + (−0.0758 + 0.131i)31-s − 0.612·35-s + 0.796·37-s + (−0.323 + 0.560i)41-s + (−0.335 − 0.581i)43-s + (0.574 + 0.994i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.199 + 0.979i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.199 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1682206490\)
\(L(\frac12)\) \(\approx\) \(0.1682206490\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.81 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.95 + 3.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.53 - 4.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.03T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (2.47 - 4.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.60 + 7.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.422 - 0.731i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.84T + 37T^{2} \)
41 \( 1 + (2.07 - 3.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 + 3.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.93 - 6.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (-5.60 + 9.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.208 + 0.360i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.05T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + (-7.56 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.932 + 1.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.669T + 89T^{2} \)
97 \( 1 + (7.63 + 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.215229498371254752780317015773, −7.79876250691657637728508762035, −7.05663871988195064036193847196, −6.33375447707313988199823008948, −5.56932722195616736309359986391, −4.47666363022448378455687515467, −3.62822571101305024571655880093, −2.88088545204763044400856591454, −2.00080065709692955821154692469, −0.05900055950328354858513398085, 1.04066176398532804186018031290, 2.29466862664853630658030902757, 3.50071967159617536914500762043, 4.43326577996077646298239326533, 5.00341350287385514507131680929, 5.55306042466904203247485693503, 6.92447348034901542887893350584, 7.66362041654099058252211538856, 8.041211413971459591250458227277, 8.850563092716404886589526768124

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