Properties

Degree $2$
Conductor $3024$
Sign $0.997 - 0.0709i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.794 + 1.37i)5-s + (−1.23 + 2.33i)7-s + (0.794 − 1.37i)11-s + (2.40 − 4.16i)13-s + (2.69 + 4.67i)17-s + (3.54 − 6.14i)19-s + (−0.150 − 0.260i)23-s + (1.23 − 2.14i)25-s + (−4.13 − 7.16i)29-s + 2.71·31-s + (−4.19 + 0.153i)35-s + (0.5 − 0.866i)37-s + (−2.93 + 5.08i)41-s + (0.833 + 1.44i)43-s + 2.66·47-s + ⋯
L(s)  = 1  + (0.355 + 0.615i)5-s + (−0.468 + 0.883i)7-s + (0.239 − 0.414i)11-s + (0.667 − 1.15i)13-s + (0.654 + 1.13i)17-s + (0.814 − 1.41i)19-s + (−0.0313 − 0.0542i)23-s + (0.247 − 0.429i)25-s + (−0.768 − 1.33i)29-s + 0.487·31-s + (−0.709 + 0.0258i)35-s + (0.0821 − 0.142i)37-s + (−0.458 + 0.794i)41-s + (0.127 + 0.220i)43-s + 0.388·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0709i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.997 - 0.0709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024209051\)
\(L(\frac12)\) \(\approx\) \(2.024209051\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 2.33i)T \)
good5 \( 1 + (-0.794 - 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.794 + 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.150 + 0.260i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.71T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.833 - 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.539556299966313131873550871178, −8.214545459215741867316455438766, −7.14300071231126727365715741543, −6.25707022200238915798787516146, −5.87601176450412189652357663133, −5.07384927799347345046122348297, −3.75974650312896350434505666452, −3.04395319062033234423858849295, −2.29756603743776511745785500798, −0.818978873605584258411545165474, 1.00000448501820043552246953722, 1.78792542907884659169624944994, 3.30555379565931250640606262639, 3.89951364282526885830926863199, 4.89427529526693304110870979904, 5.58065649839161968066325306835, 6.55744958465804086654564116965, 7.19255509243252300888514005993, 7.85744218526082067700825815448, 8.939734406277146868114770972321

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