Properties

Degree $2$
Conductor $3024$
Sign $0.562 - 0.826i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.927 − 1.60i)5-s + (−0.900 − 2.48i)7-s + (1.28 + 2.23i)11-s + (2.82 + 4.88i)13-s + (−3.57 + 6.19i)17-s + (−0.636 − 1.10i)19-s + (−0.120 + 0.208i)23-s + (0.777 + 1.34i)25-s + (−0.923 + 1.59i)29-s + 2.99·31-s + (−4.83 − 0.862i)35-s + (0.338 + 0.585i)37-s + (0.733 + 1.27i)41-s + (−4.14 + 7.17i)43-s − 12.3·47-s + ⋯
L(s)  = 1  + (0.414 − 0.718i)5-s + (−0.340 − 0.940i)7-s + (0.388 + 0.672i)11-s + (0.782 + 1.35i)13-s + (−0.868 + 1.50i)17-s + (−0.146 − 0.252i)19-s + (−0.0251 + 0.0435i)23-s + (0.155 + 0.269i)25-s + (−0.171 + 0.297i)29-s + 0.537·31-s + (−0.817 − 0.145i)35-s + (0.0556 + 0.0963i)37-s + (0.114 + 0.198i)41-s + (−0.631 + 1.09i)43-s − 1.79·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.562 - 0.826i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.562 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608303714\)
\(L(\frac12)\) \(\approx\) \(1.608303714\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.900 + 2.48i)T \)
good5 \( 1 + (-0.927 + 1.60i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.28 - 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.82 - 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.57 - 6.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.636 + 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.120 - 0.208i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.923 - 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.99T + 31T^{2} \)
37 \( 1 + (-0.338 - 0.585i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.733 - 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.14 - 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + (3.35 - 5.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 - 4.83T + 67T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 + (6.55 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.72T + 79T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.60 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.40 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.753431926192152694651174260365, −8.335565391603811135054424570890, −7.14013485594410187825395138587, −6.59836385713685411359321289694, −5.98259155879603707756453920942, −4.68067199600164510690561685297, −4.30275196620503971279290401323, −3.43007370983575660643150091827, −1.89890129288456703513724826822, −1.27353143064181166504633663862, 0.51946680389617295392925198048, 2.13120330381291050929008195289, 2.94189318130114579588356783304, 3.55876157205202768220601934093, 4.92753459657422354658721149176, 5.64563690660391279283496145683, 6.37586953450353956487209700128, 6.84896314897364535354630261652, 8.032075276518863747777332560295, 8.585486154041682777203371530223

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