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Properties

Label 2-3024-9.7-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.620 - 0.784i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 − 1.47i)5-s + (−0.5 + 0.866i)7-s + (−1.23 + 2.14i)11-s + (−0.388 − 0.673i)13-s − 2.81·17-s + 4.98·19-s + (−0.356 − 0.616i)23-s + (1.05 − 1.82i)25-s + (2.25 − 3.90i)29-s + (2.54 + 4.41i)31-s + 1.69·35-s − 6.87·37-s + (−2.93 − 5.08i)41-s + (−2.32 + 4.03i)43-s + (−6.49 + 11.2i)47-s + ⋯
L(s)  = 1  + (−0.380 − 0.658i)5-s + (−0.188 + 0.327i)7-s + (−0.373 + 0.646i)11-s + (−0.107 − 0.186i)13-s − 0.681·17-s + 1.14·19-s + (−0.0742 − 0.128i)23-s + (0.211 − 0.365i)25-s + (0.418 − 0.725i)29-s + (0.457 + 0.793i)31-s + 0.287·35-s − 1.13·37-s + (−0.458 − 0.794i)41-s + (−0.354 + 0.614i)43-s + (−0.947 + 1.64i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.620 - 0.784i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5078232679\)
\(L(\frac12)\) \(\approx\) \(0.5078232679\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.849 + 1.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.23 - 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.388 + 0.673i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 + (0.356 + 0.616i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.25 + 3.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.54 - 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.32 - 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.49 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.88T + 53T^{2} \)
59 \( 1 + (7.14 + 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 - 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 4.98T + 73T^{2} \)
79 \( 1 + (4.60 - 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + (4.32 - 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.878209126167150313238586037774, −8.312354626417637168523003017124, −7.55172650265911794323257884699, −6.78988564144371998661966752697, −5.92229898744580994539710788293, −4.93966300060479496488262284144, −4.55722254406702205510854809435, −3.37679737026773433946573289190, −2.48478581264795878982577228999, −1.26177971472684245819120758954, 0.16428096490706031055429452358, 1.66733414498183871787746634501, 3.03508962188100466643556940719, 3.41653623758742040770993786458, 4.56486924910723740355514807280, 5.36015179954075741543300332880, 6.31890010308374071753062219637, 7.00969102786693112347171336961, 7.59344047252017112920516168073, 8.438102866395140361689562154030

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