Properties

Degree $2$
Conductor $3024$
Sign $-0.801 - 0.598i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.85·5-s + (2.60 − 0.464i)7-s − 2.57·11-s + (2.82 + 4.88i)13-s + (−3.57 − 6.19i)17-s + (−0.636 + 1.10i)19-s + 0.241·23-s − 1.55·25-s + (−0.923 + 1.59i)29-s + (−1.49 + 2.59i)31-s + (−4.83 + 0.862i)35-s + (0.338 − 0.585i)37-s + (0.733 + 1.27i)41-s + (−4.14 + 7.17i)43-s + (6.15 + 10.6i)47-s + ⋯
L(s)  = 1  − 0.829·5-s + (0.984 − 0.175i)7-s − 0.776·11-s + (0.782 + 1.35i)13-s + (−0.868 − 1.50i)17-s + (−0.146 + 0.252i)19-s + 0.0503·23-s − 0.311·25-s + (−0.171 + 0.297i)29-s + (−0.268 + 0.465i)31-s + (−0.817 + 0.145i)35-s + (0.0556 − 0.0963i)37-s + (0.114 + 0.198i)41-s + (−0.631 + 1.09i)43-s + (0.898 + 1.55i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.801 - 0.598i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5296057789\)
\(L(\frac12)\) \(\approx\) \(0.5296057789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 0.464i)T \)
good5 \( 1 + 1.85T + 5T^{2} \)
11 \( 1 + 2.57T + 11T^{2} \)
13 \( 1 + (-2.82 - 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.57 + 6.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.636 - 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.241T + 23T^{2} \)
29 \( 1 + (0.923 - 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.49 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.733 - 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.14 - 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.15 - 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 + 5.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.04 - 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.47 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.86 + 3.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.60 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.40 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.952799024533502380271530354417, −8.188805562810799905945601576961, −7.57982580838838009724777686291, −6.93898899418356975976654774192, −6.02714857692776688214682091962, −4.85732573101178745726742904803, −4.51845010546147864311821136544, −3.56806453475869472084650939491, −2.44523802380740902114256087383, −1.37547014512322238719196758224, 0.16691419018284007198539028164, 1.61809908517829356769577363581, 2.69454051743092466476160838646, 3.80557902220153614018292480748, 4.37211747580997025643810485374, 5.46886093882116679534157996316, 5.91888327147531558984202386013, 7.14588035006905820805244426398, 7.81723956492293742110425676275, 8.418499970678818412696347598395

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