↑

Properties

Label 2-3024-252.31-c1-0-40
Degree $2$
Conductor $3024$
Sign $-0.609 + 0.792i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.280i·5-s + (−0.164 − 2.64i)7-s − 2.82i·11-s + (−4.06 − 2.34i)13-s + (7.00 + 4.04i)17-s + (0.474 + 0.821i)19-s + 0.392i·23-s + 4.92·25-s + (−1.51 − 2.61i)29-s + (−1.06 − 1.84i)31-s + (−0.741 + 0.0462i)35-s + (−2.43 − 4.21i)37-s + (0.478 + 0.276i)41-s + (4.28 − 2.47i)43-s + (1.39 − 2.41i)47-s + ⋯
L(s)  = 1  − 0.125i·5-s + (−0.0622 − 0.998i)7-s − 0.850i·11-s + (−1.12 − 0.651i)13-s + (1.69 + 0.980i)17-s + (0.108 + 0.188i)19-s + 0.0818i·23-s + 0.984·25-s + (−0.280 − 0.486i)29-s + (−0.191 − 0.330i)31-s + (−0.125 + 0.00781i)35-s + (−0.400 − 0.693i)37-s + (0.0748 + 0.0431i)41-s + (0.653 − 0.377i)43-s + (0.203 − 0.352i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.609 + 0.792i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.283077717\)
\(L(\frac12)\) \(\approx\) \(1.283077717\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.164 + 2.64i)T \)
good5 \( 1 + 0.280iT - 5T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (4.06 + 2.34i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-7.00 - 4.04i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.474 - 0.821i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.392iT - 23T^{2} \)
29 \( 1 + (1.51 + 2.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.06 + 1.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.43 + 4.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.478 - 0.276i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.28 + 2.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.39 + 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.70 + 9.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.67 + 5.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.85 + 4.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (0.542 + 0.313i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.3 + 7.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.30 + 9.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.90 - 5.71i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.6 - 7.86i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.235477506041032100762980194379, −7.74769990181816876273336518850, −7.13277834464456498052279764688, −6.05574195634798299293472401316, −5.47294963466743785399448421992, −4.52986277760725120165423904790, −3.59648513996819348105446595868, −2.94873535704737589115655245892, −1.49073366268667909094145214573, −0.41352609007570741073269883120, 1.42202510598987405194464259623, 2.56574971980172177669431311354, 3.15811887735455540991089059004, 4.53417601102362315345258193893, 5.11512173176275800486738347105, 5.83156861749997307913468859463, 6.98153294285996179145812993265, 7.29037788130465442356421775135, 8.258851037442985017434723201573, 9.118056974520406328518160768910

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