Properties

Degree $2$
Conductor $3024$
Sign $-0.709 - 0.705i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.918 + 1.59i)5-s + (0.361 + 2.62i)7-s + (1.54 + 2.68i)11-s + (2.40 + 4.16i)13-s + (−1.87 + 3.24i)17-s + (2.71 + 4.70i)19-s + (3.97 − 6.89i)23-s + (0.813 + 1.40i)25-s + (0.325 − 0.563i)29-s − 1.03·31-s + (−4.50 − 1.83i)35-s + (0.873 + 1.51i)37-s + (−2.52 − 4.36i)41-s + (6.09 − 10.5i)43-s − 4.61·47-s + ⋯
L(s)  = 1  + (−0.410 + 0.711i)5-s + (0.136 + 0.990i)7-s + (0.466 + 0.808i)11-s + (0.666 + 1.15i)13-s + (−0.453 + 0.786i)17-s + (0.622 + 1.07i)19-s + (0.829 − 1.43i)23-s + (0.162 + 0.281i)25-s + (0.0604 − 0.104i)29-s − 0.186·31-s + (−0.760 − 0.309i)35-s + (0.143 + 0.248i)37-s + (−0.393 − 0.682i)41-s + (0.929 − 1.61i)43-s − 0.672·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.709 - 0.705i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.709 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658107617\)
\(L(\frac12)\) \(\approx\) \(1.658107617\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.361 - 2.62i)T \)
good5 \( 1 + (0.918 - 1.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.54 - 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.97 + 6.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.325 + 0.563i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.03T + 31T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.61T + 47T^{2} \)
53 \( 1 + (4.55 - 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.79T + 59T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-3.83 + 6.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.76 - 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.985914104148701472371834947526, −8.349654753453073688017074340231, −7.43895555732994318601948568387, −6.65584563077476186695697857980, −6.19100424042304171843527919030, −5.14304487987302305784591664037, −4.22407869646262440872269797122, −3.49880398765636915222433200030, −2.39446903227212145036331763432, −1.55785306238091510382877546235, 0.59220694381835544696752582918, 1.22710298140789278767648783937, 3.00860135548033738614656685544, 3.57906064367333087731130872586, 4.65419143032891493764001280176, 5.16566915806583977964924741908, 6.20181660054651783637237949242, 7.02550827272633936057511470199, 7.80270299467403430103537890267, 8.324495950655441882705164408167

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