Properties

Degree $2$
Conductor $3024$
Sign $-0.628 - 0.777i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 2.18i)5-s + (−2.63 − 0.275i)7-s + (2.85 − 4.94i)11-s + (−2.45 + 4.24i)13-s + (−2.49 − 4.32i)17-s + (0.00383 − 0.00664i)19-s + (−0.333 − 0.578i)23-s + (−0.682 + 1.18i)25-s + (−3.85 − 6.66i)29-s + 7.76·31-s + (2.71 + 6.09i)35-s + (−3.19 + 5.53i)37-s + (−5.21 + 9.02i)41-s + (−4.42 − 7.67i)43-s + 2.16·47-s + ⋯
L(s)  = 1  + (−0.564 − 0.977i)5-s + (−0.994 − 0.104i)7-s + (0.861 − 1.49i)11-s + (−0.680 + 1.17i)13-s + (−0.605 − 1.04i)17-s + (0.000880 − 0.00152i)19-s + (−0.0696 − 0.120i)23-s + (−0.136 + 0.236i)25-s + (−0.715 − 1.23i)29-s + 1.39·31-s + (0.459 + 1.03i)35-s + (−0.525 + 0.909i)37-s + (−0.813 + 1.40i)41-s + (−0.675 − 1.16i)43-s + 0.315·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.628 - 0.777i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1348002306\)
\(L(\frac12)\) \(\approx\) \(0.1348002306\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.275i)T \)
good5 \( 1 + (1.26 + 2.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.85 + 4.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.45 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.49 + 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00383 + 0.00664i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.333 + 0.578i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.85 + 6.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 + (3.19 - 5.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.21 - 9.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.42 + 7.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + (-3.69 - 6.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.523T + 59T^{2} \)
61 \( 1 + 8.99T + 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + 5.68T + 71T^{2} \)
73 \( 1 + (1.52 + 2.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.16T + 79T^{2} \)
83 \( 1 + (0.258 + 0.448i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.19 - 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.469371891615654270467586813626, −7.49329085713159561493480856754, −6.59043548924063476524039268103, −6.17127159332887666190744216652, −4.97523966145878015161378142149, −4.31651100178169353456816603843, −3.53102735102941267361603466858, −2.54819638227765813389484447954, −1.05049082079088986351640355500, −0.04774354698359463056917147684, 1.80694016326564937193011977471, 2.88452265183293205497621316998, 3.58465562045425701876692963851, 4.38248058368188151657250405760, 5.44772787478103159005024549764, 6.40690065380131595356820072478, 7.03137103544030425037120290096, 7.41210665670210880168981846770, 8.458446634396619322571309640613, 9.258346205630195304018228416695

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