Properties

Degree $2$
Conductor $3024$
Sign $-0.694 - 0.719i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 + 2.95i)5-s + (0.410 + 2.61i)7-s + (−2.69 + 4.67i)11-s + (1.89 − 3.28i)13-s + (−0.411 − 0.713i)17-s + (−0.233 + 0.404i)19-s + (2.74 + 4.76i)23-s + (−3.30 + 5.72i)25-s + (−0.400 − 0.693i)29-s + 9.90·31-s + (−7.01 + 5.66i)35-s + (4.34 − 7.52i)37-s + (−1.84 + 3.19i)41-s + (4.36 + 7.55i)43-s − 10.4·47-s + ⋯
L(s)  = 1  + (0.761 + 1.31i)5-s + (0.155 + 0.987i)7-s + (−0.813 + 1.40i)11-s + (0.525 − 0.910i)13-s + (−0.0999 − 0.173i)17-s + (−0.0535 + 0.0928i)19-s + (0.573 + 0.993i)23-s + (−0.661 + 1.14i)25-s + (−0.0743 − 0.128i)29-s + 1.77·31-s + (−1.18 + 0.957i)35-s + (0.713 − 1.23i)37-s + (−0.288 + 0.498i)41-s + (0.665 + 1.15i)43-s − 1.53·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.694 - 0.719i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.928580798\)
\(L(\frac12)\) \(\approx\) \(1.928580798\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.410 - 2.61i)T \)
good5 \( 1 + (-1.70 - 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.69 - 4.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.89 + 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.411 + 0.713i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.233 - 0.404i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.74 - 4.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.400 + 0.693i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.90T + 31T^{2} \)
37 \( 1 + (-4.34 + 7.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.84 - 3.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.36 - 7.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + (-4.71 - 8.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.66T + 59T^{2} \)
61 \( 1 - 0.948T + 61T^{2} \)
67 \( 1 + 0.539T + 67T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 - 4.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.82T + 79T^{2} \)
83 \( 1 + (3.79 + 6.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.73 + 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.22 + 5.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.163476939745762365845198196469, −8.081786098798499906898846861587, −7.53103696297409809117114162529, −6.67651840187665776853250947147, −5.95774246465351019827370789607, −5.35588043602530673230450885541, −4.41831777345889050188693186882, −2.94403746354338513639214825356, −2.68696768171388779364842836258, −1.63253100891879675792735304875, 0.61510051897893988822713904448, 1.39772039971405404870052466781, 2.65454505835734879938998777496, 3.81253141255148297545614373668, 4.67554998311755119208411919007, 5.23029737424462019452669594409, 6.20792898120994325408041544793, 6.74021320649738840100292337476, 8.017514924795030220817269853861, 8.456222464672149754333893090235

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