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Properties

Label 2-3024-252.67-c0-0-0
Degree $2$
Conductor $3024$
Sign $0.916 + 0.400i$
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  + 5-s + (−0.866 − 0.5i)7-s + i·11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + i·23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.499 + 0.866i)49-s + (−0.5 + 0.866i)53-s + i·55-s + ⋯
L(s)  = 1  + 5-s + (−0.866 − 0.5i)7-s + i·11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + i·23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.499 + 0.866i)49-s + (−0.5 + 0.866i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (3007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.418620655\)
\(L(\frac12)\) \(\approx\) \(1.418620655\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.284857819858757035328516028725, −7.898739064990568610624159817971, −7.36550265378910738641813814531, −6.59923345317052854669072940504, −5.72985335025634246474894284283, −5.23622848318525716016595415803, −4.08011534403630535356001535356, −3.17377534971714774847937605544, −2.30700318763082098220377092855, −1.02919977329600778195907393244, 1.30117087454715303637527662235, 2.40337941388952180740836899914, 3.30950899098424419667728438376, 4.12976162721740679816401452201, 5.47457360067674072990355435016, 5.93125340005176059779962328998, 6.42547790007052301292026376083, 7.38335438218933669118819974048, 8.471090716047725031895301165943, 8.930054579106656279101457949689

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