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Properties

Label 2-3024-252.151-c0-0-0
Degree $2$
Conductor $3024$
Sign $0.235 - 0.971i$
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  + (−0.5 + 0.866i)5-s − i·7-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)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 − 49-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s − i·7-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)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 − 49-s + (−0.5 + 0.866i)53-s − 0.999i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9563314527\)
\(L(\frac12)\) \(\approx\) \(0.9563314527\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
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 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
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

−9.130680432760841210372895568614, −8.086396591920560564232303262272, −7.45528737412635707382265075085, −6.98648285350276958251176295666, −6.23855556048393868464401854942, −5.08918332368591696083888510317, −4.32801871187835299889323082051, −3.48432609803354060336713541422, −2.71036382798636382182032483222, −1.36332432913943315222377182796, 0.62954673643284491079388875588, 2.16103565836499962778557059869, 3.06400758832735126725693764875, 4.08448671174729871691021361369, 4.95118348678505293455503965954, 5.71821361189017655058669172472, 6.18982039825063660518910486887, 7.53896398665500997442963493818, 8.094218136886009120720873222336, 8.773725739093007103610078879660

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