Properties

Degree $2$
Conductor $3024$
Sign $-0.579 - 0.814i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 2.31i)5-s + (−0.581 − 2.58i)7-s + (−0.682 + 1.18i)11-s + (−2.75 + 4.77i)13-s + (1.23 + 2.14i)17-s + (2.19 − 3.80i)19-s + (2.34 + 4.06i)23-s + (−1.05 + 1.83i)25-s + (−2.94 − 5.10i)29-s − 3.11·31-s + (5.18 − 4.78i)35-s + (−3.15 + 5.46i)37-s + (−1.38 + 2.40i)41-s + (4.87 + 8.45i)43-s − 10.0·47-s + ⋯
L(s)  = 1  + (0.596 + 1.03i)5-s + (−0.219 − 0.975i)7-s + (−0.205 + 0.356i)11-s + (−0.764 + 1.32i)13-s + (0.300 + 0.520i)17-s + (0.503 − 0.872i)19-s + (0.488 + 0.846i)23-s + (−0.211 + 0.366i)25-s + (−0.547 − 0.948i)29-s − 0.559·31-s + (0.877 − 0.809i)35-s + (−0.518 + 0.898i)37-s + (−0.216 + 0.375i)41-s + (0.744 + 1.28i)43-s − 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.579 - 0.814i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.579 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267583182\)
\(L(\frac12)\) \(\approx\) \(1.267583182\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.581 + 2.58i)T \)
good5 \( 1 + (-1.33 - 2.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.75 - 4.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.23 - 2.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.19 + 3.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.34 - 4.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.94 + 5.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.11T + 31T^{2} \)
37 \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.38 - 2.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 8.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (-1.47 - 2.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.55T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 + 8.29T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.11 + 1.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + (-5.15 - 8.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.73 - 13.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.55 + 4.42i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.314542869924263319188302716049, −8.041429785512737603567459993368, −7.25136511944887082538940634988, −6.85292068613647829773260370682, −6.16906993751779257103955456633, −5.08496389946251228699718247530, −4.28427426124649796407915952941, −3.33887562874485420143445751072, −2.46987844001277422286297414051, −1.44471555125280190272201085556, 0.38160215509133737901614256665, 1.68603578340428136214318679872, 2.71584519697591607003224589904, 3.53934170048792393072348070362, 4.95968025564493667740934831280, 5.41017512758756767933005028238, 5.80639870343907328399972722449, 7.01800266878837032331893128018, 7.81973056923527748117577056319, 8.663617493232305867412811999287

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