↑

Properties

Label 2-3024-252.31-c1-0-35
Degree $2$
Conductor $3024$
Sign $-0.110 + 0.993i$
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  − 1.47i·5-s + (2.64 − 0.0201i)7-s + 2.77i·11-s + (−0.955 − 0.551i)13-s + (−1.69 − 0.978i)17-s + (−3.46 − 6.00i)19-s − 0.0322i·23-s + 2.81·25-s + (−4.53 − 7.86i)29-s + (0.352 + 0.609i)31-s + (−0.0297 − 3.90i)35-s + (−1.92 − 3.33i)37-s + (−2.23 − 1.29i)41-s + (8.95 − 5.17i)43-s + (−2.14 + 3.70i)47-s + ⋯
L(s)  = 1  − 0.660i·5-s + (0.999 − 0.00761i)7-s + 0.836i·11-s + (−0.264 − 0.152i)13-s + (−0.410 − 0.237i)17-s + (−0.794 − 1.37i)19-s − 0.00673i·23-s + 0.563·25-s + (−0.842 − 1.45i)29-s + (0.0632 + 0.109i)31-s + (−0.00502 − 0.660i)35-s + (−0.316 − 0.548i)37-s + (−0.349 − 0.201i)41-s + (1.36 − 0.788i)43-s + (−0.312 + 0.540i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.593369252\)
\(L(\frac12)\) \(\approx\) \(1.593369252\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0201i)T \)
good5 \( 1 + 1.47iT - 5T^{2} \)
11 \( 1 - 2.77iT - 11T^{2} \)
13 \( 1 + (0.955 + 0.551i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.69 + 0.978i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.46 + 6.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.0322iT - 23T^{2} \)
29 \( 1 + (4.53 + 7.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.352 - 0.609i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.92 + 3.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.23 + 1.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.95 + 5.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.14 - 3.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.49 - 6.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.568 + 0.985i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.24 - 1.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 5.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (4.41 + 2.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.26 - 4.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.34 + 9.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.83 - 5.10i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.95 - 4.01i)T + (48.5 - 84.0i)T^{2} \)
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

−8.636061997112189665104559434718, −7.71840229448882976150436845689, −7.20437223028309569585520030036, −6.24057727401177163592415425946, −5.21050143941958269407924170742, −4.67547346663821626644082237920, −4.07301355986862081754835133698, −2.56915983178245977052766079924, −1.84059541628733855120641868441, −0.50254662218129644968762773268, 1.33456961961242083301891170170, 2.31186283734703189772169932472, 3.36754876845496471801989897831, 4.16828958970365914519905435203, 5.14046368708901305316081440384, 5.87405936001405129537502562291, 6.71392654808923153145452076691, 7.40172123499171528220894740043, 8.369096811076500429389814118167, 8.595349320186706180117734773675

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