↑

Properties

Label 2-3024-63.59-c1-0-10
Degree $2$
Conductor $3024$
Sign $-0.205 - 0.978i$
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  + 0.623·5-s + (0.996 − 2.45i)7-s + 5.19i·11-s + (−2.74 − 1.58i)13-s + (−0.437 + 0.757i)17-s + (1.41 − 0.819i)19-s + 8.25i·23-s − 4.61·25-s + (−4.96 + 2.86i)29-s + (−4.02 + 2.32i)31-s + (0.621 − 1.52i)35-s + (1.24 + 2.15i)37-s + (3.52 − 6.10i)41-s + (1.56 + 2.70i)43-s + (−4.73 + 8.19i)47-s + ⋯
L(s)  = 1  + 0.278·5-s + (0.376 − 0.926i)7-s + 1.56i·11-s + (−0.760 − 0.438i)13-s + (−0.106 + 0.183i)17-s + (0.325 − 0.187i)19-s + 1.72i·23-s − 0.922·25-s + (−0.921 + 0.532i)29-s + (−0.722 + 0.417i)31-s + (0.105 − 0.258i)35-s + (0.204 + 0.353i)37-s + (0.550 − 0.953i)41-s + (0.238 + 0.413i)43-s + (−0.690 + 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.205 - 0.978i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188259488\)
\(L(\frac12)\) \(\approx\) \(1.188259488\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.996 + 2.45i)T \)
good5 \( 1 - 0.623T + 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (2.74 + 1.58i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.437 - 0.757i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 0.819i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.25iT - 23T^{2} \)
29 \( 1 + (4.96 - 2.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.02 - 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.24 - 2.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.52 + 6.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.56 - 2.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.15 - 0.665i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.18 + 5.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.65 - 5.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.04 - 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (11.6 + 6.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.84 + 8.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.192 + 0.332i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0198 + 0.0344i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.94 + 3.43i)T + (48.5 - 84.0i)T^{2} \)
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

−9.135797154070738448420651715253, −7.894967470506104098274318232233, −7.39667519857186188190654670915, −6.99978292660947111318270561787, −5.74856298315319014543099691073, −5.07471091496886258207726388216, −4.28281639474847742027862669346, −3.46222139994211299652193832107, −2.18526830353427181210008828577, −1.37654401228583343410155115719, 0.35711884030762115996719728707, 1.94131216311834256563512322580, 2.66218978186004823922944429609, 3.70904263306742733098385537186, 4.71154137217642026197350061655, 5.61302974954562489331429993793, 6.02966770714595169202271512820, 6.95282335879029710717263448201, 7.984735581549517609702732907058, 8.451032803998869319504728326617

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