Properties

Degree $2$
Conductor $3024$
Sign $0.342 - 0.939i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 2.40i)5-s + (−1.74 + 1.99i)7-s + (1.71 + 2.97i)11-s + (−0.429 − 0.743i)13-s + (0.405 − 0.701i)17-s + (−0.750 − 1.29i)19-s + (−3.82 + 6.62i)23-s + (−1.34 − 2.32i)25-s + (−3.99 + 6.92i)29-s + 7.21·31-s + (2.37 + 6.93i)35-s + (0.458 + 0.793i)37-s + (−1.67 − 2.90i)41-s + (−1.20 + 2.08i)43-s − 0.615·47-s + ⋯
L(s)  = 1  + (0.619 − 1.07i)5-s + (−0.657 + 0.753i)7-s + (0.518 + 0.898i)11-s + (−0.119 − 0.206i)13-s + (0.0982 − 0.170i)17-s + (−0.172 − 0.298i)19-s + (−0.797 + 1.38i)23-s + (−0.268 − 0.464i)25-s + (−0.742 + 1.28i)29-s + 1.29·31-s + (0.400 + 1.17i)35-s + (0.0753 + 0.130i)37-s + (−0.261 − 0.453i)41-s + (−0.183 + 0.318i)43-s − 0.0897·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.342 - 0.939i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.498586564\)
\(L(\frac12)\) \(\approx\) \(1.498586564\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 - 1.99i)T \)
good5 \( 1 + (-1.38 + 2.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.71 - 2.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.429 + 0.743i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.405 + 0.701i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.750 + 1.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 - 6.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.99 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.21T + 31T^{2} \)
37 \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.67 + 2.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.20 - 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.615T + 47T^{2} \)
53 \( 1 + (6.31 - 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 16.2T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 + (5.75 - 9.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.11 - 8.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 6.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.000105860201259805031747426610, −8.299045759808004872043690063062, −7.33140903011815058917111904033, −6.53952782536461046010541492442, −5.68179551954295624380830536661, −5.16231745395900958140402797074, −4.28639876982408335806222531070, −3.22965102775261567887532191730, −2.13348756939333852498971317794, −1.24968301077428451542708388082, 0.47876148693572942442738323906, 1.99650007185679929103850457163, 2.94012929774291993700014374589, 3.71485392797936179615917392745, 4.52586563364829033536645487733, 5.89958638108933681353196615341, 6.36133382764194959865994230213, 6.80900849189602743362774566985, 7.82273574245503573011339920217, 8.513670164934714583771437476252

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