Properties

Degree $2$
Conductor $3024$
Sign $-0.427 + 0.903i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.52·5-s + (−1.16 − 2.37i)7-s − 2.32·11-s + (−2.35 − 4.08i)13-s + (0.636 + 1.10i)17-s + (−2.78 + 4.82i)19-s − 3.29·23-s + 7.45·25-s + (4.32 − 7.48i)29-s + (4.25 − 7.37i)31-s + (−4.11 − 8.38i)35-s + (−2.84 + 4.91i)37-s + (−1.66 − 2.88i)41-s + (−0.0444 + 0.0769i)43-s + (−3.52 − 6.10i)47-s + ⋯
L(s)  = 1  + 1.57·5-s + (−0.441 − 0.897i)7-s − 0.699·11-s + (−0.654 − 1.13i)13-s + (0.154 + 0.267i)17-s + (−0.638 + 1.10i)19-s − 0.687·23-s + 1.49·25-s + (0.802 − 1.38i)29-s + (0.764 − 1.32i)31-s + (−0.696 − 1.41i)35-s + (−0.466 + 0.808i)37-s + (−0.260 − 0.450i)41-s + (−0.00677 + 0.0117i)43-s + (−0.514 − 0.890i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.427 + 0.903i$
Motivic weight: \(1\)
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.427 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.539966976\)
\(L(\frac12)\) \(\approx\) \(1.539966976\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 + 2.37i)T \)
good5 \( 1 - 3.52T + 5T^{2} \)
11 \( 1 + 2.32T + 11T^{2} \)
13 \( 1 + (2.35 + 4.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.636 - 1.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.78 - 4.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 + (-4.32 + 7.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.25 + 7.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.84 - 4.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.66 + 2.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0444 - 0.0769i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.52 + 6.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 + 5.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.99 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.67 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.06 + 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.01 + 8.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.90 - 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.561 - 0.972i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.50 - 6.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.141861304236288015389999325230, −8.038515729211181269302405825179, −6.77707642776162231071982724606, −6.16749964858267006533253204628, −5.55260429487550105409068655667, −4.71406326877444688441697743600, −3.64081843181894398366276809335, −2.62431568398869237818611566029, −1.83737970626761934713331562405, −0.43262646761847895930406992725, 1.55101937992909563228321142779, 2.46236856537826677103155593703, 2.95536415018285733104564618500, 4.60068845211600179786393714060, 5.15189329711910496805116316671, 5.96574713807662239008801924931, 6.59687593224323750634637280299, 7.22606761569637296717973636727, 8.574755323038108093883582086435, 8.981265144504915445174806842053

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