Properties

Label 2-3024-63.41-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.0329 - 0.999i$
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  + (−1.60 + 2.77i)5-s + (0.555 + 2.58i)7-s + (4.13 − 2.38i)11-s + (−0.861 − 0.497i)13-s + 6.51·17-s + 5.38i·19-s + (6.56 + 3.79i)23-s + (−2.63 − 4.57i)25-s + (2.93 − 1.69i)29-s + (−3.51 − 2.02i)31-s + (−8.07 − 2.60i)35-s + 6.29·37-s + (3.48 − 6.04i)41-s + (−3.81 − 6.60i)43-s + (3.78 + 6.54i)47-s + ⋯
L(s)  = 1  + (−0.716 + 1.24i)5-s + (0.209 + 0.977i)7-s + (1.24 − 0.720i)11-s + (−0.239 − 0.137i)13-s + 1.57·17-s + 1.23i·19-s + (1.36 + 0.790i)23-s + (−0.527 − 0.914i)25-s + (0.545 − 0.314i)29-s + (−0.630 − 0.364i)31-s + (−1.36 − 0.440i)35-s + 1.03·37-s + (0.544 − 0.943i)41-s + (−0.581 − 1.00i)43-s + (0.551 + 0.955i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.0329 - 0.999i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.0329 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.910140801\)
\(L(\frac12)\) \(\approx\) \(1.910140801\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.555 - 2.58i)T \)
good5 \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.13 + 2.38i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.861 + 0.497i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 - 5.38iT - 19T^{2} \)
23 \( 1 + (-6.56 - 3.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.93 + 1.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.51 + 2.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.29T + 37T^{2} \)
41 \( 1 + (-3.48 + 6.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.81 + 6.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.78 - 6.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.77iT - 53T^{2} \)
59 \( 1 + (2.25 - 3.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.26 + 3.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.493 - 0.854i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.20iT - 71T^{2} \)
73 \( 1 + 2.63iT - 73T^{2} \)
79 \( 1 + (-7.96 - 13.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.49 + 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.42T + 89T^{2} \)
97 \( 1 + (3.13 - 1.81i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.913447010303153305463146421031, −7.997580938173038002764938716527, −7.51268922985648567104042487576, −6.63181058649371230726613628759, −5.90510212100480592358308625491, −5.26142632920320436826987515171, −3.82266986971222491058636709166, −3.42916267081378726979633807639, −2.52237375444111247059971817296, −1.17704959527257574718809504436, 0.78203282807360366522355016134, 1.34714676546090019263244830575, 3.00907287507089266583553483937, 4.02474291141208208232032701232, 4.62621057434683008659159193360, 5.11247048078503504549005948649, 6.44386883776161111995638283277, 7.17085989163124385436305386331, 7.74007877653147365530458387760, 8.577291788975010783831615775496

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