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Properties

Label 2-3024-63.41-c1-0-0
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: $\chi_{3024} (881, \cdot )$
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

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

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