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Properties

Label 2-3024-63.47-c1-0-15
Degree $2$
Conductor $3024$
Sign $0.447 - 0.894i$
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  − 2.59·5-s + (2.61 + 0.417i)7-s + 3.54i·11-s + (2.74 − 1.58i)13-s + (0.487 + 0.844i)17-s + (−2.11 − 1.21i)19-s − 3.37i·23-s + 1.72·25-s + (−0.267 − 0.154i)29-s + (4.35 + 2.51i)31-s + (−6.77 − 1.08i)35-s + (−3.47 + 6.01i)37-s + (6.08 + 10.5i)41-s + (5.47 − 9.48i)43-s + (−1.43 − 2.48i)47-s + ⋯
L(s)  = 1  − 1.15·5-s + (0.987 + 0.157i)7-s + 1.06i·11-s + (0.760 − 0.438i)13-s + (0.118 + 0.204i)17-s + (−0.484 − 0.279i)19-s − 0.704i·23-s + 0.345·25-s + (−0.0497 − 0.0286i)29-s + (0.783 + 0.452i)31-s + (−1.14 − 0.183i)35-s + (−0.570 + 0.988i)37-s + (0.949 + 1.64i)41-s + (0.835 − 1.44i)43-s + (−0.209 − 0.362i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.447 - 0.894i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.469000483\)
\(L(\frac12)\) \(\approx\) \(1.469000483\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 - 0.417i)T \)
good5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 3.54iT - 11T^{2} \)
13 \( 1 + (-2.74 + 1.58i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.487 - 0.844i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.11 + 1.21i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.37iT - 23T^{2} \)
29 \( 1 + (0.267 + 0.154i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.35 - 2.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.47 - 6.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.08 - 10.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.47 + 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.43 + 2.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.81 - 4.51i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.219 - 0.380i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.41 + 1.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.82 + 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.25iT - 71T^{2} \)
73 \( 1 + (14.0 - 8.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.49 - 6.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.31 - 4.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.4 - 7.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.471308465679335535347860410933, −8.246546088159583586519572329234, −7.46863682880253510117965146166, −6.74808325295536992037965270373, −5.78085238785230910712604970654, −4.67948666292571934484840389627, −4.37257580185399922123658384365, −3.34539524753428769661149652912, −2.23064245844347646344514116158, −1.05383127517184571438504461047, 0.56210156301098697223155413466, 1.74693780064702830138420362731, 3.10407388045284458501359527888, 3.95638238153627264334393956200, 4.46754009851904760208814235376, 5.58142648708055663160249143984, 6.23948835138488319083695955028, 7.42419929039847009421730941104, 7.76174290013233259645979295240, 8.585289838930304179806350033286

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