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Properties

Label 2-3024-63.47-c1-0-24
Degree $2$
Conductor $3024$
Sign $0.742 - 0.669i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.84·5-s + (2.64 + 0.0704i)7-s + 2.45i·11-s + (−3.06 + 1.76i)13-s + (−2.91 − 5.05i)17-s + (2.90 + 1.67i)19-s + 8.01i·23-s + 3.10·25-s + (1.45 + 0.839i)29-s + (3.45 + 1.99i)31-s + (7.53 + 0.200i)35-s + (4.07 − 7.06i)37-s + (5.43 + 9.41i)41-s + (−3.27 + 5.67i)43-s + (3.31 + 5.74i)47-s + ⋯
L(s)  = 1  + 1.27·5-s + (0.999 + 0.0266i)7-s + 0.738i·11-s + (−0.848 + 0.490i)13-s + (−0.708 − 1.22i)17-s + (0.665 + 0.384i)19-s + 1.67i·23-s + 0.621·25-s + (0.269 + 0.155i)29-s + (0.620 + 0.357i)31-s + (1.27 + 0.0339i)35-s + (0.670 − 1.16i)37-s + (0.849 + 1.47i)41-s + (−0.499 + 0.865i)43-s + (0.483 + 0.837i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.573818332\)
\(L(\frac12)\) \(\approx\) \(2.573818332\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.0704i)T \)
good5 \( 1 - 2.84T + 5T^{2} \)
11 \( 1 - 2.45iT - 11T^{2} \)
13 \( 1 + (3.06 - 1.76i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.90 - 1.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.01iT - 23T^{2} \)
29 \( 1 + (-1.45 - 0.839i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.45 - 1.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.43 - 9.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.27 - 5.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.31 - 5.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.64 - 4.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.178 - 0.309i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.52 + 1.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.14 + 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-5.42 + 3.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.75 + 9.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.189 + 0.327i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.05 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.00 + 2.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.142185512708813488707472012886, −7.84443175453849738110078738738, −7.43308832553730080671309781454, −6.52893930836065111580011651199, −5.66185457687464634950584994752, −4.94188973142559669912369569493, −4.43232288856939568703526466513, −2.91922970228612048801151823555, −2.09556832369024257276946074993, −1.32548059901833207386146590800, 0.839935068911233984068866724181, 2.08113895965552766356018423141, 2.63568536867394581922051526756, 4.01586294614826246015756720844, 4.91977160419720760409581273234, 5.55035995359538890686765026489, 6.29160806051651422441039908452, 7.01815074517501130616246839576, 8.168579507071453189736845209412, 8.480401874324195988952070287431

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