Properties

Label 2-3024-63.20-c1-0-43
Degree $2$
Conductor $3024$
Sign $-0.739 - 0.673i$
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  + (−1.79 − 3.10i)5-s + (2.56 + 0.634i)7-s + (−0.200 − 0.115i)11-s + (1.16 − 0.673i)13-s − 7.94·17-s − 3.06i·19-s + (−4.87 + 2.81i)23-s + (−3.91 + 6.78i)25-s + (2.33 + 1.34i)29-s + (−1.85 + 1.07i)31-s + (−2.63 − 9.10i)35-s − 7.27·37-s + (0.813 + 1.40i)41-s + (0.927 − 1.60i)43-s + (−0.0396 + 0.0686i)47-s + ⋯
L(s)  = 1  + (−0.801 − 1.38i)5-s + (0.970 + 0.239i)7-s + (−0.0604 − 0.0348i)11-s + (0.323 − 0.186i)13-s − 1.92·17-s − 0.702i·19-s + (−1.01 + 0.586i)23-s + (−0.783 + 1.35i)25-s + (0.433 + 0.250i)29-s + (−0.332 + 0.192i)31-s + (−0.445 − 1.53i)35-s − 1.19·37-s + (0.127 + 0.220i)41-s + (0.141 − 0.245i)43-s + (−0.00578 + 0.0100i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.739 - 0.673i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03636604145\)
\(L(\frac12)\) \(\approx\) \(0.03636604145\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.634i)T \)
good5 \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.200 + 0.115i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 0.673i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.94T + 17T^{2} \)
19 \( 1 + 3.06iT - 19T^{2} \)
23 \( 1 + (4.87 - 2.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.33 - 1.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.85 - 1.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.27T + 37T^{2} \)
41 \( 1 + (-0.813 - 1.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.927 + 1.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0396 - 0.0686i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + (6.48 + 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.729 - 0.420i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.05 - 8.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.47iT - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 + (-3.30 + 5.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.41 + 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.56T + 89T^{2} \)
97 \( 1 + (13.1 + 7.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.366995743291697575028955987324, −7.73466811533872887513984326624, −6.85992671008239566929976550087, −5.80296408907730597840850396016, −4.92024402663796434683190560058, −4.52297090109554227376793909711, −3.70231326682026382942684069120, −2.27546095256377950034543159290, −1.30137787318036536654456526203, −0.01133854323153378115119647969, 1.84233969783852432490687877820, 2.65027649954987705062798511874, 3.87123807450169305286071488573, 4.22357757623777762530787154218, 5.33226692366055192283408745260, 6.53472842924795224865873450692, 6.78126791612008552142747553840, 7.79852267875711366568991728439, 8.201340138303362424698122218005, 9.041260031637739255833426035817

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