↑

Properties

Label 2-3024-63.20-c1-0-10
Degree $2$
Conductor $3024$
Sign $-0.919 - 0.392i$
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  + (1.94 + 3.36i)5-s + (−2.09 + 1.60i)7-s + (3.41 + 1.97i)11-s + (−2.46 + 1.42i)13-s + 0.742·17-s + 1.78i·19-s + (−5.41 + 3.12i)23-s + (−5.07 + 8.78i)25-s + (2.50 + 1.44i)29-s + (3.04 − 1.75i)31-s + (−9.50 − 3.94i)35-s + 3.00·37-s + (−5.24 − 9.08i)41-s + (−0.471 + 0.816i)43-s + (−1.09 + 1.89i)47-s + ⋯
L(s)  = 1  + (0.870 + 1.50i)5-s + (−0.793 + 0.608i)7-s + (1.03 + 0.594i)11-s + (−0.684 + 0.395i)13-s + 0.179·17-s + 0.409i·19-s + (−1.12 + 0.651i)23-s + (−1.01 + 1.75i)25-s + (0.464 + 0.268i)29-s + (0.546 − 0.315i)31-s + (−1.60 − 0.666i)35-s + 0.493·37-s + (−0.819 − 1.41i)41-s + (−0.0719 + 0.124i)43-s + (−0.159 + 0.276i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.919 - 0.392i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570343462\)
\(L(\frac12)\) \(\approx\) \(1.570343462\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 1.60i)T \)
good5 \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.41 - 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.46 - 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.742T + 17T^{2} \)
19 \( 1 - 1.78iT - 19T^{2} \)
23 \( 1 + (5.41 - 3.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.50 - 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.04 + 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.471 - 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.09 - 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.0105 + 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.72 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 - 4.85iT - 73T^{2} \)
79 \( 1 + (-1.81 + 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.02 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 + (16.2 + 9.40i)T + (48.5 + 84.0i)T^{2} \)
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

−9.341960726086527014402870293328, −8.339804871654989694992583482355, −7.15868251724916325193439125902, −6.85641636325486366327912485662, −6.07700104745558341908074466023, −5.54936585842149662024149556713, −4.20331399004081927172217984165, −3.34859636268635467610922127080, −2.48856691481003390346175710844, −1.77732936110195526991086183092, 0.48468326016984530984498979122, 1.36271619458429914144653747618, 2.58717729090257642753735862583, 3.73907593013133091484184660271, 4.56649613313736623638845148962, 5.25743077518233595008634940979, 6.25890229150994903380618272389, 6.56295427037108398867699179159, 7.85524216307380429341993207724, 8.465799403849910692383407808246

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