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Properties

Label 2-3024-63.41-c1-0-27
Degree $2$
Conductor $3024$
Sign $0.927 + 0.374i$
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  + (−1.94 + 3.36i)5-s + (−0.343 − 2.62i)7-s + (3.41 − 1.97i)11-s + (2.46 + 1.42i)13-s − 0.742·17-s + 1.78i·19-s + (−5.41 − 3.12i)23-s + (−5.07 − 8.78i)25-s + (2.50 − 1.44i)29-s + (−3.04 − 1.75i)31-s + (9.50 + 3.94i)35-s + 3.00·37-s + (5.24 − 9.08i)41-s + (−0.471 − 0.816i)43-s + (1.09 + 1.89i)47-s + ⋯
L(s)  = 1  + (−0.870 + 1.50i)5-s + (−0.130 − 0.991i)7-s + (1.03 − 0.594i)11-s + (0.684 + 0.395i)13-s − 0.179·17-s + 0.409i·19-s + (−1.12 − 0.651i)23-s + (−1.01 − 1.75i)25-s + (0.464 − 0.268i)29-s + (−0.546 − 0.315i)31-s + (1.60 + 0.666i)35-s + 0.493·37-s + (0.819 − 1.41i)41-s + (−0.0719 − 0.124i)43-s + (0.159 + 0.276i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417199174\)
\(L(\frac12)\) \(\approx\) \(1.417199174\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.343 + 2.62i)T \)
good5 \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.41 + 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.742T + 17T^{2} \)
19 \( 1 - 1.78iT - 19T^{2} \)
23 \( 1 + (5.41 + 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.04 + 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.471 + 0.816i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.09 - 1.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.0105 + 0.0183i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.13 - 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.72 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 - 4.85iT - 73T^{2} \)
79 \( 1 + (-1.81 - 3.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 + (-16.2 + 9.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.546546976130961406060120881088, −7.76158143181207523650364883169, −7.19767980921652417630341332395, −6.38953643906804642195941013148, −6.08462609614207045346176939423, −4.38230025860496450857619928890, −3.82550136186454894656213830938, −3.33245551890390988830679409024, −2.08729356707212144061518106666, −0.57844109585857873780466125014, 0.940668564392303221984320349764, 1.92898227771708857488957879302, 3.32593501492497098048005274747, 4.17017787098006297062273941773, 4.81120610861855516321651907945, 5.66190622662033486288077213930, 6.37918845903955129519934329715, 7.45774097487911555243409538492, 8.204362358349742864846474096942, 8.750204222705407468102082430969

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