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Properties

Label 2-3024-336.149-c0-0-3
Degree $2$
Conductor $3024$
Sign $0.440 + 0.897i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + i·7-s + (0.707 − 0.707i)8-s + (−0.499 − 0.866i)10-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (−0.366 − 1.36i)19-s + (−0.707 − 0.707i)20-s + (0.707 − 1.22i)23-s + (0.499 + 0.866i)28-s + (−0.707 − 0.707i)29-s + (1 + 1.73i)31-s + (0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + i·7-s + (0.707 − 0.707i)8-s + (−0.499 − 0.866i)10-s + (0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (−0.366 − 1.36i)19-s + (−0.707 − 0.707i)20-s + (0.707 − 1.22i)23-s + (0.499 + 0.866i)28-s + (−0.707 − 0.707i)29-s + (1 + 1.73i)31-s + (0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.233747020\)
\(L(\frac12)\) \(\approx\) \(2.233747020\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.820554246124661567801668936455, −8.126236400042378265092960095452, −7.01903178546460152865849227980, −6.37424716844004219578472881642, −5.49550549690150569480297350845, −4.78107485376453954176987599796, −4.36877910953126269727427383003, −3.03518764318064512441326084884, −2.40752488645712053058207063247, −1.10277731143920598696381122967, 1.64217398508412690628041009443, 2.83324131999761806573908414646, 3.69219961305865987664064297977, 4.09857645867846031231385829428, 5.25388281912344893119747262377, 6.00839445526147704700650788866, 6.80789645126637331040190962969, 7.42977153239614019715665642991, 7.81106504338710993159311528247, 8.914946825992202229717346126901

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