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Properties

Label 2-3024-63.20-c1-0-14
Degree $2$
Conductor $3024$
Sign $0.703 - 0.710i$
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  + (0.266 + 0.462i)5-s + (1.89 + 1.84i)7-s + (−3.39 − 1.96i)11-s + (0.116 − 0.0674i)13-s + 4.32·17-s + 2.22i·19-s + (−1.70 + 0.983i)23-s + (2.35 − 4.08i)25-s + (5.16 + 2.98i)29-s + (−0.800 + 0.462i)31-s + (−0.348 + 1.36i)35-s + 7.79·37-s + (−4.59 − 7.95i)41-s + (−3.24 + 5.62i)43-s + (3.04 − 5.27i)47-s + ⋯
L(s)  = 1  + (0.119 + 0.206i)5-s + (0.715 + 0.698i)7-s + (−1.02 − 0.591i)11-s + (0.0324 − 0.0187i)13-s + 1.04·17-s + 0.511i·19-s + (−0.355 + 0.205i)23-s + (0.471 − 0.816i)25-s + (0.959 + 0.553i)29-s + (−0.143 + 0.0829i)31-s + (−0.0589 + 0.231i)35-s + 1.28·37-s + (−0.716 − 1.24i)41-s + (−0.494 + 0.857i)43-s + (0.443 − 0.768i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.703 - 0.710i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.950156620\)
\(L(\frac12)\) \(\approx\) \(1.950156620\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 - 1.84i)T \)
good5 \( 1 + (-0.266 - 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.39 + 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.116 + 0.0674i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + (1.70 - 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.16 - 2.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.800 - 0.462i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (4.59 + 7.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.04 + 5.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (-1.89 - 3.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.35 - 5.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.75 - 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 + 0.381iT - 73T^{2} \)
79 \( 1 + (-4.60 + 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.28 + 2.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + (13.6 + 7.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.601556237392717832730523645989, −8.152087131129965682362071328374, −7.49719834619883159892151608140, −6.45363020667072599551251547789, −5.60888888655008434624990591308, −5.20119005202002201349444858296, −4.12832563841265494199841679414, −3.02165426639307271176325781729, −2.33781078907667066273145933426, −1.05748802766411680906821475772, 0.73188380009712035298654417598, 1.87933232143127857629306052779, 2.92455384823617942062105396607, 3.97582046953404620530228822152, 4.91866195529537339067847126454, 5.28074630961741370482408524634, 6.45621909515080696210130225006, 7.21838205581833216568076182605, 8.020805290805689644545345198268, 8.303326868474161641016718604608

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