↑

Properties

Label 2-3024-63.20-c1-0-4
Degree $2$
Conductor $3024$
Sign $-0.919 - 0.393i$
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  + (−0.183 − 0.317i)5-s + (0.624 + 2.57i)7-s + (0.579 + 0.334i)11-s + (0.867 − 0.500i)13-s − 4.98·17-s + 6.35i·19-s + (−6.66 + 3.84i)23-s + (2.43 − 4.21i)25-s + (−1.58 − 0.914i)29-s + (5.47 − 3.16i)31-s + (0.701 − 0.669i)35-s − 5.16·37-s + (−2.15 − 3.73i)41-s + (−2.24 + 3.89i)43-s + (−4.16 + 7.21i)47-s + ⋯
L(s)  = 1  + (−0.0819 − 0.141i)5-s + (0.235 + 0.971i)7-s + (0.174 + 0.100i)11-s + (0.240 − 0.138i)13-s − 1.21·17-s + 1.45i·19-s + (−1.38 + 0.802i)23-s + (0.486 − 0.842i)25-s + (−0.294 − 0.169i)29-s + (0.983 − 0.568i)31-s + (0.118 − 0.113i)35-s − 0.849·37-s + (−0.337 − 0.584i)41-s + (−0.343 + 0.594i)43-s + (−0.607 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.919 - 0.393i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.919 - 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6795888217\)
\(L(\frac12)\) \(\approx\) \(0.6795888217\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.624 - 2.57i)T \)
good5 \( 1 + (0.183 + 0.317i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.579 - 0.334i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.867 + 0.500i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.98T + 17T^{2} \)
19 \( 1 - 6.35iT - 19T^{2} \)
23 \( 1 + (6.66 - 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 + 0.914i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.47 + 3.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + (2.15 + 3.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.16 - 7.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (4.36 + 7.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.29 - 2.47i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.44 + 9.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 - 4.07iT - 73T^{2} \)
79 \( 1 + (-4.17 + 7.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.50 - 14.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (-14.9 - 8.60i)T + (48.5 + 84.0i)T^{2} \)
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

−9.035733233189304621668819258963, −8.135489821314239396043305533058, −7.918302565595786062206673033793, −6.54053312005172325549150154121, −6.11444593811114706347971026207, −5.24487430158582148620997276989, −4.38321834970331355294703833712, −3.54538615559759238165524932067, −2.38646932085188765359424836871, −1.59669708982763601941658103714, 0.20304761126139529250283161600, 1.53368837545615769247711270293, 2.66689908668796882824658386635, 3.71541031086018596940919408397, 4.46886228266810059427287777002, 5.13414272331218182884683311064, 6.40712842033486350090324763273, 6.83424173057523529464525053212, 7.53667067048648315688035219928, 8.571822908638368716840173250798

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