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Properties

Label 2-3024-63.16-c1-0-31
Degree $2$
Conductor $3024$
Sign $0.666 + 0.745i$
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  + 2.92·5-s + (−2.35 + 1.20i)7-s − 1.35·11-s + (−0.733 − 1.26i)13-s + (−1.65 − 2.86i)17-s + (1.10 − 1.91i)19-s + 2.62·23-s + 3.53·25-s + (−0.521 + 0.903i)29-s + (1.63 − 2.83i)31-s + (−6.88 + 3.50i)35-s + (5.43 − 9.41i)37-s + (0.904 + 1.56i)41-s + (2.17 − 3.76i)43-s + (−1.98 − 3.44i)47-s + ⋯
L(s)  = 1  + 1.30·5-s + (−0.891 + 0.453i)7-s − 0.408·11-s + (−0.203 − 0.352i)13-s + (−0.401 − 0.695i)17-s + (0.253 − 0.438i)19-s + 0.548·23-s + 0.706·25-s + (−0.0968 + 0.167i)29-s + (0.294 − 0.509i)31-s + (−1.16 + 0.592i)35-s + (0.893 − 1.54i)37-s + (0.141 + 0.244i)41-s + (0.331 − 0.573i)43-s + (−0.290 − 0.502i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.666 + 0.745i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.872032773\)
\(L(\frac12)\) \(\approx\) \(1.872032773\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 - 1.20i)T \)
good5 \( 1 - 2.92T + 5T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 + (0.733 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 + 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + (0.521 - 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.904 - 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.17 + 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.22 - 5.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.279 + 0.484i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.40 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.383 - 0.664i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.20 - 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.14 - 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.875891586773592820641490715972, −7.85646081021663927684834884162, −6.93923123319206375054032393186, −6.34182080176383874681582729406, −5.51056042755134480766368661379, −5.08177796756113088643288668334, −3.76807685200328637106733673811, −2.65435651339171379543195578666, −2.23590598055761209791304804532, −0.62798176126628509347487860874, 1.13150407851056161255324622823, 2.24835361005609727056687976287, 3.07371308301303623628734298017, 4.12089109530361986970828463940, 5.06870642710924113013695346045, 5.92943557159358586135923520801, 6.45431878442781906691755826861, 7.15032713106349849472861020272, 8.135074480648343760237877820579, 8.955553380797680067869825812950

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