Properties

Degree $2$
Conductor $3024$
Sign $0.841 + 0.540i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.170 + 0.294i)5-s + (2.63 + 0.253i)7-s + (0.335 − 0.581i)11-s + (1.62 − 2.81i)13-s + (1.10 + 1.90i)17-s + (−0.242 + 0.419i)19-s + (−2.09 − 3.62i)23-s + (2.44 − 4.22i)25-s + (−0.478 − 0.829i)29-s − 2.08·31-s + (0.373 + 0.818i)35-s + (4.81 − 8.34i)37-s + (3.90 − 6.75i)41-s + (3.66 + 6.34i)43-s − 2.69·47-s + ⋯
L(s)  = 1  + (0.0760 + 0.131i)5-s + (0.995 + 0.0957i)7-s + (0.101 − 0.175i)11-s + (0.450 − 0.779i)13-s + (0.266 + 0.462i)17-s + (−0.0555 + 0.0961i)19-s + (−0.436 − 0.756i)23-s + (0.488 − 0.845i)25-s + (−0.0889 − 0.154i)29-s − 0.374·31-s + (0.0631 + 0.138i)35-s + (0.791 − 1.37i)37-s + (0.609 − 1.05i)41-s + (0.558 + 0.967i)43-s − 0.393·47-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: $0.841 + 0.540i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.171448146\)
\(L(\frac12)\) \(\approx\) \(2.171448146\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.253i)T \)
good5 \( 1 + (-0.170 - 0.294i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.335 + 0.581i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.62 + 2.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.10 - 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.242 - 0.419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.478 + 0.829i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.90 + 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.66 - 6.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.69T + 47T^{2} \)
53 \( 1 + (-6.12 - 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.94T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + 5.57T + 71T^{2} \)
73 \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (-2.47 - 4.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.52 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.23 - 7.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.643654762314034896814929133714, −7.85899355584563619591919082047, −7.39761103644331514531641347907, −6.09728963078297308879352301099, −5.81612898313312804006214362444, −4.68826582780616278140675540131, −4.04230423312380097838115741802, −2.92162903695747218407586795098, −1.97519380541405544892339993946, −0.78816974346501774074050969235, 1.16226843212450879236798022919, 1.98260675826460585894278022018, 3.21003184062844321997499719520, 4.19731631976208092658473037307, 4.89557278059742663160073155405, 5.65580641112635116936408482547, 6.57339129090151110513861991874, 7.39438653258154869917684135547, 7.984370559739577949726029678788, 8.850874485984596782159661189748

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