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Properties

Label 2-3024-63.4-c1-0-18
Degree $2$
Conductor $3024$
Sign $0.902 - 0.430i$
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  − 5-s + (2.5 − 0.866i)7-s − 3·11-s + (−0.5 + 0.866i)13-s + (1.5 − 2.59i)17-s + (2.5 + 4.33i)19-s + 23-s − 4·25-s + (4.5 + 7.79i)29-s + (2 + 3.46i)31-s + (−2.5 + 0.866i)35-s + (−2.5 − 4.33i)37-s + (3.5 − 6.06i)41-s + (1.5 + 2.59i)43-s + (−4 + 6.92i)47-s + ⋯
L(s)  = 1  − 0.447·5-s + (0.944 − 0.327i)7-s − 0.904·11-s + (−0.138 + 0.240i)13-s + (0.363 − 0.630i)17-s + (0.573 + 0.993i)19-s + 0.208·23-s − 0.800·25-s + (0.835 + 1.44i)29-s + (0.359 + 0.622i)31-s + (−0.422 + 0.146i)35-s + (−0.410 − 0.711i)37-s + (0.546 − 0.946i)41-s + (0.228 + 0.396i)43-s + (−0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.708529099\)
\(L(\frac12)\) \(\approx\) \(1.708529099\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.5 - 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.630761824680766008392743097752, −7.88044711960424269751298579745, −7.51120869595981274591099048053, −6.66020524652141385923430568603, −5.43145873188991103697542389766, −5.05363163419361489263026114983, −4.06903613412630640982372646917, −3.21591403942139127077247268277, −2.10620975065855239124172583419, −0.945935363060354378818826789773, 0.68810538194226671814781024992, 2.08897423254132417888772836327, 2.91209152286232268447401669629, 4.03904077272981341553028983678, 4.85055526992447312414220306792, 5.47794211455982396839471683781, 6.36515592126686927332800768735, 7.39043881051625499766336514430, 8.073123532876619268877100683468, 8.321698046885026295542377701516

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