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Properties

Label 2-3024-63.16-c1-0-45
Degree $2$
Conductor $3024$
Sign $-0.970 - 0.242i$
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  + 0.146·5-s + (−0.0802 − 2.64i)7-s + 1.66·11-s + (0.0999 + 0.173i)13-s + (−3.13 − 5.43i)17-s + (−3.45 + 5.99i)19-s − 6.18·23-s − 4.97·25-s + (2.46 − 4.27i)29-s + (−1.25 + 2.18i)31-s + (−0.0117 − 0.386i)35-s + (−3.50 + 6.06i)37-s + (−1.15 − 2.00i)41-s + (0.940 − 1.62i)43-s + (0.905 + 1.56i)47-s + ⋯
L(s)  = 1  + 0.0654·5-s + (−0.0303 − 0.999i)7-s + 0.501·11-s + (0.0277 + 0.0480i)13-s + (−0.760 − 1.31i)17-s + (−0.793 + 1.37i)19-s − 1.28·23-s − 0.995·25-s + (0.458 − 0.793i)29-s + (−0.226 + 0.391i)31-s + (−0.00198 − 0.0653i)35-s + (−0.575 + 0.996i)37-s + (−0.180 − 0.313i)41-s + (0.143 − 0.248i)43-s + (0.132 + 0.228i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.970 - 0.242i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1405907206\)
\(L(\frac12)\) \(\approx\) \(0.1405907206\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0802 + 2.64i)T \)
good5 \( 1 - 0.146T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 + (-0.0999 - 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 + (-2.46 + 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 - 2.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.15 + 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.940 + 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.905 - 1.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.28 + 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.339 - 0.587i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.09 - 5.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (0.778 + 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.39 - 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.75 + 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.53 - 7.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.98 - 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.199077374865613570512632872236, −7.63534499694658329903874959373, −6.73644507769096566522285070749, −6.21726821280336647652316668226, −5.19840252213055952659567308629, −4.16382512385435844729581904463, −3.80481402237314715258090550715, −2.47743919365645954774835629937, −1.44082017598333008549847657392, −0.04128997251212310244700253950, 1.78744672425327192745689331314, 2.43451851104359993856837978129, 3.67323064617030304609262186550, 4.40335818833104457723619116594, 5.39587464798286907303272622751, 6.18013707017509222281774920819, 6.65502655077854676351134359067, 7.72891770147748721491304575744, 8.592706509378418830162480865648, 8.924749930112492297646466946004

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