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Properties

Label 2-3024-252.223-c1-0-20
Degree $2$
Conductor $3024$
Sign $0.999 + 0.0372i$
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  + (−2.10 + 1.21i)5-s + (1.48 − 2.18i)7-s + (1.03 + 0.596i)11-s + (1.39 − 0.805i)13-s + 4.15i·17-s + 0.184·19-s + (2.92 − 1.68i)23-s + (0.447 − 0.775i)25-s + (1.87 − 3.25i)29-s + (−3.73 − 6.47i)31-s + (−0.472 + 6.40i)35-s − 0.585·37-s + (−8.51 + 4.91i)41-s + (4.32 + 2.49i)43-s + (−0.418 + 0.725i)47-s + ⋯
L(s)  = 1  + (−0.940 + 0.542i)5-s + (0.562 − 0.826i)7-s + (0.311 + 0.179i)11-s + (0.386 − 0.223i)13-s + 1.00i·17-s + 0.0423·19-s + (0.609 − 0.351i)23-s + (0.0895 − 0.155i)25-s + (0.348 − 0.604i)29-s + (−0.671 − 1.16i)31-s + (−0.0799 + 1.08i)35-s − 0.0962·37-s + (−1.32 + 0.767i)41-s + (0.659 + 0.380i)43-s + (−0.0611 + 0.105i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0372i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.999 + 0.0372i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.618653365\)
\(L(\frac12)\) \(\approx\) \(1.618653365\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.18i)T \)
good5 \( 1 + (2.10 - 1.21i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.03 - 0.596i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.39 + 0.805i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.15iT - 17T^{2} \)
19 \( 1 - 0.184T + 19T^{2} \)
23 \( 1 + (-2.92 + 1.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.87 + 3.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.73 + 6.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.585T + 37T^{2} \)
41 \( 1 + (8.51 - 4.91i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.32 - 2.49i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.418 - 0.725i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.65T + 53T^{2} \)
59 \( 1 + (-2.02 - 3.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.99 - 1.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.62 + 5.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + (-13.7 - 7.95i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.42 + 5.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (3.27 + 1.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.442200853911488912716853490220, −7.981976292097081330807550880203, −7.26378372837485298668424678723, −6.63814692331931098584921177378, −5.71590574034261524071470102206, −4.61438497195026570826561593189, −3.94857783231959256391408157474, −3.33619376014851263688906080903, −2.01343730545336308004564516725, −0.76605477001918640867954987205, 0.812435099278733195941671280659, 2.01622207160772083472674046425, 3.20741516900049374107440608205, 3.99035394928766211539077677315, 5.03737839016474284978705110517, 5.34922889066993252500961153464, 6.59185103046768395150292277647, 7.27261110793114879816710110417, 8.109121340625138894091256630344, 8.809885490646691814993567626929

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