↑

Properties

Label 2-3024-63.20-c1-0-33
Degree $2$
Conductor $3024$
Sign $0.995 - 0.0987i$
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  + (0.717 + 1.24i)5-s + (2.40 + 1.11i)7-s + (2.80 + 1.61i)11-s + (4.43 − 2.55i)13-s − 1.09·17-s − 4.48i·19-s + (3.47 − 2.00i)23-s + (1.47 − 2.54i)25-s + (−1.02 − 0.593i)29-s + (3.24 − 1.87i)31-s + (0.342 + 3.77i)35-s − 0.239·37-s + (−3.71 − 6.43i)41-s + (3.82 − 6.62i)43-s + (−2.11 + 3.65i)47-s + ⋯
L(s)  = 1  + (0.320 + 0.555i)5-s + (0.907 + 0.419i)7-s + (0.844 + 0.487i)11-s + (1.22 − 0.709i)13-s − 0.264·17-s − 1.02i·19-s + (0.723 − 0.417i)23-s + (0.294 − 0.509i)25-s + (−0.191 − 0.110i)29-s + (0.582 − 0.336i)31-s + (0.0579 + 0.638i)35-s − 0.0393·37-s + (−0.580 − 1.00i)41-s + (0.583 − 1.00i)43-s + (−0.307 + 0.533i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.995 - 0.0987i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.995 - 0.0987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.580810359\)
\(L(\frac12)\) \(\approx\) \(2.580810359\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.40 - 1.11i)T \)
good5 \( 1 + (-0.717 - 1.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.80 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
23 \( 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.02 + 0.593i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 1.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.239T + 37T^{2} \)
41 \( 1 + (3.71 + 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.82 + 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.11 - 3.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.01iT - 53T^{2} \)
59 \( 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.330 - 0.571i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 - 7.31iT - 73T^{2} \)
79 \( 1 + (-1.83 + 3.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.45 - 9.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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

−8.764866454071919897037750163808, −8.087553224625070492320238212683, −7.12917736768117172055758930988, −6.49639861301685900149410036409, −5.74630714180428381054524659144, −4.85892019426149490619075791828, −4.07326618049789866219880942922, −2.98624900379650701282643882264, −2.11749777674361968329802076107, −1.00382157066688748475923176260, 1.23924887444796631326962584354, 1.59155995350040717721511957836, 3.22567809360792357106409113391, 4.06414879121867590095155956965, 4.77341893711350736574563664356, 5.68552122801998477977136676649, 6.39065985179766612765949962090, 7.19131990687570700280450580931, 8.170818167138502549689567477190, 8.677519553216842229721963469262

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