Properties

Degree $2$
Conductor $3024$
Sign $-0.819 + 0.572i$
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 + (−2.54 − 0.714i)7-s + (1.99 − 3.45i)11-s + (1.00 − 1.73i)13-s + (3.57 + 6.18i)17-s + (4.01 − 6.96i)19-s + (0.443 + 0.768i)23-s + (−1.06 + 1.83i)25-s + (1.35 + 2.33i)29-s + 1.22·31-s + (1.74 + 6.84i)35-s + (5.26 − 9.11i)37-s + (1.43 − 2.48i)41-s + (−3.40 − 5.88i)43-s − 12.1·47-s + ⋯
L(s)  = 1  + (−0.596 − 1.03i)5-s + (−0.962 − 0.270i)7-s + (0.600 − 1.04i)11-s + (0.277 − 0.480i)13-s + (0.866 + 1.50i)17-s + (0.922 − 1.59i)19-s + (0.0925 + 0.160i)23-s + (−0.212 + 0.367i)25-s + (0.250 + 0.434i)29-s + 0.220·31-s + (0.295 + 1.15i)35-s + (0.865 − 1.49i)37-s + (0.224 − 0.388i)41-s + (−0.518 − 0.898i)43-s − 1.77·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.819 + 0.572i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.819 + 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.200400598\)
\(L(\frac12)\) \(\approx\) \(1.200400598\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.714i)T \)
good5 \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.99 + 3.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.00 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.57 - 6.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.01 + 6.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.443 - 0.768i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.35 - 2.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + (-5.26 + 9.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.43 + 2.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.40 + 5.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + (-2.38 - 4.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 + 9.49T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 4.62T + 71T^{2} \)
73 \( 1 + (-2.01 - 3.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 + (5.26 + 9.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.72 - 2.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.12 + 1.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.568530897912581659161435760885, −7.75915839398659639326010312469, −6.94602636413409397612292031374, −6.03368811449838631029918773213, −5.45467491219616157850711968343, −4.38953700441206842770977923569, −3.62609689661854136180827204304, −2.98817568044914477730022063558, −1.24205553038133971121418862352, −0.43945456997306706304981626426, 1.37317897308729464656783617868, 2.81457100600281517868004130892, 3.29655298675411585219750799984, 4.19363784922152459447477904668, 5.18575351035813744103845137257, 6.28915643849842807593957883771, 6.71298699318243574456832206778, 7.48970693115427481141807841247, 8.045773487395996107429399857562, 9.245865245510771632902392941905

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