↑

Properties

Label 2-3024-9.4-c1-0-0
Degree $2$
Conductor $3024$
Sign $-0.607 - 0.794i$
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  + (1.23 − 2.13i)5-s + (−0.5 − 0.866i)7-s + (−2.32 − 4.02i)11-s + (−3.55 + 6.15i)13-s + 4.51·17-s − 4.32·19-s + (−2.93 + 5.08i)23-s + (−0.527 − 0.912i)25-s + (−3.48 − 6.04i)29-s + (−3.69 + 6.39i)31-s − 2.46·35-s − 0.726·37-s + (0.136 − 0.236i)41-s + (−2.41 − 4.18i)43-s + (−1.83 − 3.18i)47-s + ⋯
L(s)  = 1  + (0.550 − 0.952i)5-s + (−0.188 − 0.327i)7-s + (−0.700 − 1.21i)11-s + (−0.985 + 1.70i)13-s + 1.09·17-s − 0.992·19-s + (−0.611 + 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.647 − 1.12i)29-s + (−0.662 + 1.14i)31-s − 0.415·35-s − 0.119·37-s + (0.0213 − 0.0369i)41-s + (−0.368 − 0.638i)43-s + (−0.267 − 0.463i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2647329511\)
\(L(\frac12)\) \(\approx\) \(0.2647329511\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.32 + 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (2.93 - 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.48 + 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.726T + 37T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.41 + 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.83 + 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.05T + 53T^{2} \)
59 \( 1 + (4.56 - 7.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 4.32T + 73T^{2} \)
79 \( 1 + (-3.21 - 5.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.742 + 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.016910258928424303580711814544, −8.343266966271059273494864531699, −7.52723493637594909994205662826, −6.73404383960902498845380438145, −5.74625748233041512131205476119, −5.29570317442192403352244746394, −4.33534996739385558363689224654, −3.51756582843446544938489524001, −2.28056531545647835145056872750, −1.36092471068714041722190054377, 0.07703309323769526066293749891, 2.01637869856489855829989560073, 2.64151666004063412786011658146, 3.44139948088772122480365129184, 4.74885580426786280591138429617, 5.40587280562642887970046517966, 6.19171641373999061706259424491, 6.95517671439638668460136017319, 7.76770042319494897497644646729, 8.180353208158076537078661002565

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