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Properties

Label 2-3024-63.25-c1-0-18
Degree $2$
Conductor $3024$
Sign $0.888 - 0.458i$
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  + (−0.5 + 0.866i)5-s + (2 − 1.73i)7-s + (−1.5 − 2.59i)11-s + (−1.5 − 2.59i)13-s + (−2.5 + 4.33i)17-s + (3.5 + 6.06i)19-s + (−2.5 + 4.33i)23-s + (2 + 3.46i)25-s + (−0.5 + 0.866i)29-s + 8·31-s + (0.499 + 2.59i)35-s + (−1.5 − 2.59i)37-s + (−2.5 − 4.33i)41-s + (−3.5 + 6.06i)43-s + 8·47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.452 − 0.783i)11-s + (−0.416 − 0.720i)13-s + (−0.606 + 1.05i)17-s + (0.802 + 1.39i)19-s + (−0.521 + 0.902i)23-s + (0.400 + 0.692i)25-s + (−0.0928 + 0.160i)29-s + 1.43·31-s + (0.0845 + 0.439i)35-s + (−0.246 − 0.427i)37-s + (−0.390 − 0.676i)41-s + (−0.533 + 0.924i)43-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.739491106\)
\(L(\frac12)\) \(\approx\) \(1.739491106\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.5 - 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.5 + 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.502126454974946977156568558113, −7.976246581333499161615268397809, −7.49947951790350752871941780633, −6.53469120398822558067806825849, −5.64810537893280040054241938327, −5.02947479739200056244636824909, −3.86115613184622424019639688676, −3.36336220260422682396450800960, −2.10483438988000501002280541050, −0.956328442503161389105120773882, 0.68751336309077482850425511667, 2.21270655737837010246126026546, 2.66978963491451597199452009829, 4.24963016207591388497818211913, 4.85289383248391443922588119454, 5.26421376901298918813373602893, 6.65329169333129619240583794778, 7.03831425123102573437737671373, 8.113512167687631832112314405937, 8.541105906698889280719730663060

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