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Properties

Label 2-3024-336.221-c0-0-2
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: $\chi_{3024} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ -0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7587246391\)
\(L(\frac12)\) \(\approx\) \(0.7587246391\)
\(L(1)\) 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

−8.665234545548696146403829083278, −7.956689006221134475646490906266, −7.55516551356015061818725073850, −6.35500601639789865221739571212, −5.95167114313690893565857705316, −4.49268177431870610384665387523, −4.03507021819180344919711341584, −2.73660975480310330487908834705, −1.66812580235461954340306089309, −0.66184400862286512551204337731, 1.50706660547553309075909611246, 2.66520510591178828963953080102, 3.03670903627529021534646618706, 4.70796154788230108928376870299, 5.61079371552891968511357523961, 6.37128273830474586443176237957, 6.86250563980677270536906593811, 7.67000234493290457704561843438, 8.536693788434088051133304573737, 9.055647105398606498944457496932

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