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Properties

Label 2-3024-252.31-c1-0-7
Degree $2$
Conductor $3024$
Sign $-0.841 - 0.540i$
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  + 2.81i·5-s + (1.84 + 1.89i)7-s + 2.64i·11-s + (−4.59 − 2.65i)13-s + (2.35 + 1.36i)17-s + (−0.274 − 0.474i)19-s − 2.28i·23-s − 2.90·25-s + (3.72 + 6.45i)29-s + (1.38 + 2.39i)31-s + (−5.32 + 5.19i)35-s + (1.81 + 3.13i)37-s + (7.12 + 4.11i)41-s + (−3.93 + 2.27i)43-s + (−5.61 + 9.72i)47-s + ⋯
L(s)  = 1  + 1.25i·5-s + (0.697 + 0.716i)7-s + 0.797i·11-s + (−1.27 − 0.735i)13-s + (0.571 + 0.329i)17-s + (−0.0628 − 0.108i)19-s − 0.476i·23-s − 0.581·25-s + (0.692 + 1.19i)29-s + (0.248 + 0.430i)31-s + (−0.900 + 0.877i)35-s + (0.298 + 0.516i)37-s + (1.11 + 0.642i)41-s + (−0.600 + 0.346i)43-s + (−0.818 + 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.841 - 0.540i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453575132\)
\(L(\frac12)\) \(\approx\) \(1.453575132\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 - 1.89i)T \)
good5 \( 1 - 2.81iT - 5T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 + (4.59 + 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.35 - 1.36i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.274 + 0.474i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.28iT - 23T^{2} \)
29 \( 1 + (-3.72 - 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.81 - 3.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.12 - 4.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.93 - 2.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.61 - 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.53 + 4.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.67 + 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.4 + 7.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.39 - 5.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-4.29 - 2.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.86 + 2.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.719 + 1.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.24 - 1.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.7 + 9.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.089711231399518531053146045380, −8.014441636087455162050738481731, −7.63995898440245642794128120065, −6.77340789924552777545235924804, −6.11983102581164218133695131530, −5.06376758088181011379023532384, −4.57699720551511820792238085216, −3.08245882629350215811562915812, −2.70710716436142233723106595760, −1.58635669038033285239441641287, 0.45802395500098152035308242392, 1.42741883395922435794047210458, 2.56776692925290395455012039354, 3.91463592538281626658185779313, 4.54444694817893158799555187929, 5.20473047991292282103088978244, 5.97542930004543797326272870883, 7.12106749329877829524293053388, 7.73202555954244585848320809107, 8.395942027716574288136579941245

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