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Properties

Label 2-3024-63.20-c1-0-0
Degree $2$
Conductor $3024$
Sign $0.389 - 0.920i$
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  + (−1.79 − 3.10i)5-s + (−2.25 − 1.38i)7-s + (−4.38 − 2.53i)11-s + (−5.48 + 3.16i)13-s − 0.256·17-s − 6.71i·19-s + (0.138 − 0.0802i)23-s + (−3.92 + 6.80i)25-s + (1.17 + 0.675i)29-s + (−3.01 + 1.74i)31-s + (−0.251 + 9.48i)35-s + 9.37·37-s + (−2.31 − 4.01i)41-s + (2.70 − 4.68i)43-s + (−1.50 + 2.60i)47-s + ⋯
L(s)  = 1  + (−0.801 − 1.38i)5-s + (−0.852 − 0.522i)7-s + (−1.32 − 0.763i)11-s + (−1.52 + 0.879i)13-s − 0.0622·17-s − 1.54i·19-s + (0.0289 − 0.0167i)23-s + (−0.785 + 1.36i)25-s + (0.217 + 0.125i)29-s + (−0.541 + 0.312i)31-s + (−0.0424 + 1.60i)35-s + 1.54·37-s + (−0.362 − 0.627i)41-s + (0.412 − 0.714i)43-s + (−0.219 + 0.380i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.389 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02901242046\)
\(L(\frac12)\) \(\approx\) \(0.02901242046\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
good5 \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.38 + 2.53i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.48 - 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.256T + 17T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (-0.138 + 0.0802i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.17 - 0.675i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.01 - 1.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.37T + 37T^{2} \)
41 \( 1 + (2.31 + 4.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.70 + 4.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.50 - 2.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.35iT - 53T^{2} \)
59 \( 1 + (2.32 + 4.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.50 - 1.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.52 + 7.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.74iT - 71T^{2} \)
73 \( 1 + 7.77iT - 73T^{2} \)
79 \( 1 + (6.10 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.90 + 3.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.79T + 89T^{2} \)
97 \( 1 + (-11.0 - 6.35i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.074597728481205640604037999640, −8.028959252255108286244408265766, −7.47502096886323061020831790467, −6.77975461720263477743783026316, −5.63886181765484365945605293025, −4.80561926922342424748994427725, −4.41989162214170986759653393871, −3.27951486500300615553848802597, −2.36594941650796135741022499428, −0.69168654012722653767264365004, 0.01361556604799845901496728984, 2.35141952908555113939747920795, 2.80932041509408225425277198877, 3.61142126401360990467815028814, 4.68628430500495891744718541563, 5.63364253165035958106835468145, 6.34047456450056988957436364958, 7.38275122421526715610251946664, 7.56565306364454143094969965605, 8.314200252844203697359577438879

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