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Properties

Label 2-3024-63.47-c1-0-25
Degree $2$
Conductor $3024$
Sign $-0.163 + 0.986i$
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  − 2.22·5-s + (−2.51 + 0.814i)7-s + 2.12i·11-s + (−5.10 + 2.94i)13-s + (2.34 + 4.05i)17-s + (4.54 + 2.62i)19-s − 4.36i·23-s − 0.0346·25-s + (2.25 + 1.30i)29-s + (−6.59 − 3.80i)31-s + (5.60 − 1.81i)35-s + (−1.80 + 3.12i)37-s + (−0.0395 − 0.0684i)41-s + (1.24 − 2.16i)43-s + (1.89 + 3.28i)47-s + ⋯
L(s)  = 1  − 0.996·5-s + (−0.951 + 0.307i)7-s + 0.639i·11-s + (−1.41 + 0.817i)13-s + (0.567 + 0.983i)17-s + (1.04 + 0.602i)19-s − 0.909i·23-s − 0.00693·25-s + (0.418 + 0.241i)29-s + (−1.18 − 0.683i)31-s + (0.948 − 0.306i)35-s + (−0.296 + 0.513i)37-s + (−0.00616 − 0.0106i)41-s + (0.190 − 0.329i)43-s + (0.276 + 0.479i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.163 + 0.986i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.163 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2715199004\)
\(L(\frac12)\) \(\approx\) \(0.2715199004\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 - 0.814i)T \)
good5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 - 2.12iT - 11T^{2} \)
13 \( 1 + (5.10 - 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.36iT - 23T^{2} \)
29 \( 1 + (-2.25 - 1.30i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.59 + 3.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.80 - 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0395 + 0.0684i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.24 + 2.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.89 - 3.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.08 + 2.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.59 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 - 4.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 - 4.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (12.6 - 7.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.27 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.41 + 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.9 + 7.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.504018429564903711563503065602, −7.56234286441002381643376325400, −7.22572833625610483427931379652, −6.31765939809471642470652390131, −5.44163578220922827816709422201, −4.47961960581632365238071130514, −3.78672594676562125609397000771, −2.91848290188704513306452493236, −1.83285634181005953246841509279, −0.11249319377115268536486142653, 0.850640108176246910611149781369, 2.74861903160370444163701167437, 3.25326827497400137899011795913, 4.09702298060933163850407300132, 5.20773405793496018408072198298, 5.69671233289364290929354817565, 7.11082410203033263940346637234, 7.27280596088831549268766447602, 7.982282929847019954300324417305, 9.039398194106763691795592294480

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