↑

Properties

Label 2-3024-63.41-c1-0-23
Degree $2$
Conductor $3024$
Sign $0.992 - 0.124i$
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  + (−0.0868 + 0.150i)5-s + (−1.72 + 2.00i)7-s + (3.25 − 1.87i)11-s + (3.54 + 2.04i)13-s − 6.00·17-s − 6.26i·19-s + (−6.37 − 3.68i)23-s + (2.48 + 4.30i)25-s + (8.58 − 4.95i)29-s + (3.76 + 2.17i)31-s + (−0.152 − 0.433i)35-s + 7.23·37-s + (−0.489 + 0.848i)41-s + (−0.0468 − 0.0811i)43-s + (1.86 + 3.22i)47-s + ⋯
L(s)  = 1  + (−0.0388 + 0.0672i)5-s + (−0.650 + 0.759i)7-s + (0.980 − 0.566i)11-s + (0.982 + 0.567i)13-s − 1.45·17-s − 1.43i·19-s + (−1.32 − 0.767i)23-s + (0.496 + 0.860i)25-s + (1.59 − 0.920i)29-s + (0.676 + 0.390i)31-s + (−0.0257 − 0.0732i)35-s + 1.18·37-s + (−0.0765 + 0.132i)41-s + (−0.00714 − 0.0123i)43-s + (0.271 + 0.470i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747009715\)
\(L(\frac12)\) \(\approx\) \(1.747009715\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 2.00i)T \)
good5 \( 1 + (0.0868 - 0.150i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.25 + 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.54 - 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.00T + 17T^{2} \)
19 \( 1 + 6.26iT - 19T^{2} \)
23 \( 1 + (6.37 + 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.58 + 4.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.76 - 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 + (0.489 - 0.848i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0468 + 0.0811i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.86 - 3.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.61iT - 53T^{2} \)
59 \( 1 + (0.620 - 1.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.45 + 4.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.21 - 7.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 15.4iT - 73T^{2} \)
79 \( 1 + (-1.21 - 2.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.16 + 5.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (5.02 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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

−8.806481588526033535383182786594, −8.283806876604074022532964556760, −6.95702747935992572803220111754, −6.35647033882767846835885213196, −6.06329952044680542362404325217, −4.68840357852181009378436554002, −4.10175028818177471099991727468, −3.00821695974410804530816839475, −2.23673006122487052949392922537, −0.804191538711248917643405236032, 0.826580358837102923857096462137, 1.94425294954670374102878165923, 3.24095762256912792241748946258, 4.03246810056088070637413382989, 4.55515494869136638910334341666, 5.95804405111401226428976812518, 6.39305807154864887821487943710, 7.08399751019926671652567924134, 8.087195139756034261137451734048, 8.590217060039538492637700921291

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