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Properties

Label 2-3024-252.103-c1-0-11
Degree $2$
Conductor $3024$
Sign $0.281 - 0.959i$
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  + (−0.5 − 2.59i)7-s + (3 + 1.73i)11-s + (−1.5 − 0.866i)13-s + (−1.5 + 0.866i)17-s + (−2.5 + 4.33i)19-s + (−3 + 1.73i)23-s + (−2.5 + 4.33i)25-s + (1.5 + 2.59i)29-s + 31-s + (−3.5 + 6.06i)37-s + (1.5 + 0.866i)41-s + (1.5 − 0.866i)43-s + 9·47-s + (−6.5 + 2.59i)49-s + (−4.5 − 7.79i)53-s + ⋯
L(s)  = 1  + (−0.188 − 0.981i)7-s + (0.904 + 0.522i)11-s + (−0.416 − 0.240i)13-s + (−0.363 + 0.210i)17-s + (−0.573 + 0.993i)19-s + (−0.625 + 0.361i)23-s + (−0.5 + 0.866i)25-s + (0.278 + 0.482i)29-s + 0.179·31-s + (−0.575 + 0.996i)37-s + (0.234 + 0.135i)41-s + (0.228 − 0.132i)43-s + 1.31·47-s + (−0.928 + 0.371i)49-s + (−0.618 − 1.07i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.295613678\)
\(L(\frac12)\) \(\approx\) \(1.295613678\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 15T + 59T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.863303641710569400758869881855, −8.072806085652691647947362662841, −7.30521783709530839709076233291, −6.72140558858466756734405577353, −5.93355277986632314330356637683, −4.93375262417516206642840198396, −4.02081157878349148111615025652, −3.55682627541920020372288068575, −2.15660099687245580252773285461, −1.16337622353428264424480523494, 0.43129282618097889744324184456, 2.05660924737912003910928759973, 2.70504336890518614108390483869, 3.90491175832975606772390203475, 4.61268337361711598924353731833, 5.64353420449175431426600555651, 6.29202745347741459106146260346, 6.91250975668837177835250841705, 7.927817999045557620103650863461, 8.766564795737429063828239305259

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