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Properties

Label 2-3024-63.20-c1-0-36
Degree $2$
Conductor $3024$
Sign $0.821 + 0.569i$
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  + (1.41 + 2.45i)5-s + (0.387 − 2.61i)7-s + (0.136 + 0.0789i)11-s + (3.41 − 1.97i)13-s + 4.14·17-s − 6.33i·19-s + (0.472 − 0.273i)23-s + (−1.52 + 2.64i)25-s + (−4.02 − 2.32i)29-s + (0.112 − 0.0647i)31-s + (6.98 − 2.76i)35-s − 2.46·37-s + (−1.99 − 3.45i)41-s + (−3.28 + 5.68i)43-s + (4.33 − 7.50i)47-s + ⋯
L(s)  = 1  + (0.634 + 1.09i)5-s + (0.146 − 0.989i)7-s + (0.0412 + 0.0237i)11-s + (0.947 − 0.546i)13-s + 1.00·17-s − 1.45i·19-s + (0.0986 − 0.0569i)23-s + (−0.305 + 0.528i)25-s + (−0.747 − 0.431i)29-s + (0.0201 − 0.0116i)31-s + (1.18 − 0.466i)35-s − 0.404·37-s + (−0.311 − 0.539i)41-s + (−0.500 + 0.867i)43-s + (0.632 − 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.821 + 0.569i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.821 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.209727914\)
\(L(\frac12)\) \(\approx\) \(2.209727914\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.387 + 2.61i)T \)
good5 \( 1 + (-1.41 - 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.136 - 0.0789i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.41 + 1.97i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + 6.33iT - 19T^{2} \)
23 \( 1 + (-0.472 + 0.273i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.02 + 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.112 + 0.0647i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 + (1.99 + 3.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.28 - 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.33 + 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.60iT - 53T^{2} \)
59 \( 1 + (1.80 + 3.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.91 - 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 + (-2.16 + 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.22 - 5.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.05T + 89T^{2} \)
97 \( 1 + (-2.18 - 1.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.621187687245062225426925137649, −7.75338469146884919600488045619, −7.08884115288844163984632964261, −6.48840424985744149372453777412, −5.70683676216318023113180167212, −4.81530739765765834081570557088, −3.68060302190239812620909954859, −3.11394253526044839018975045581, −2.00370923672690134813321446349, −0.75231540209878275862809687481, 1.28155776529699517088390609436, 1.87086456545859408101536880701, 3.20488393641334914399691470462, 4.12156860385273879936907110394, 5.17729468739873665692260695854, 5.68073110299594195354248822394, 6.22681317052474911936336041591, 7.40042169682117969724189374286, 8.363083693386543480808144341785, 8.693084652424487337271064425155

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