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Properties

Label 2-3024-63.16-c1-0-41
Degree $2$
Conductor $3024$
Sign $-0.855 + 0.517i$
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  − 1.33·5-s + (2.54 + 0.728i)7-s + 1.51·11-s + (−2.58 − 4.48i)13-s + (−0.774 − 1.34i)17-s + (1.25 − 2.16i)19-s − 7.36·23-s − 3.21·25-s + (0.0309 − 0.0536i)29-s + (−1.92 + 3.33i)31-s + (−3.39 − 0.972i)35-s + (−0.281 + 0.487i)37-s + (−4.51 − 7.81i)41-s + (−5.09 + 8.83i)43-s + (4.75 + 8.24i)47-s + ⋯
L(s)  = 1  − 0.596·5-s + (0.961 + 0.275i)7-s + 0.456·11-s + (−0.717 − 1.24i)13-s + (−0.187 − 0.325i)17-s + (0.287 − 0.497i)19-s − 1.53·23-s − 0.643·25-s + (0.00575 − 0.00996i)29-s + (−0.345 + 0.598i)31-s + (−0.573 − 0.164i)35-s + (−0.0462 + 0.0801i)37-s + (−0.704 − 1.22i)41-s + (−0.777 + 1.34i)43-s + (0.694 + 1.20i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.855 + 0.517i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5654189635\)
\(L(\frac12)\) \(\approx\) \(0.5654189635\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 - 0.728i)T \)
good5 \( 1 + 1.33T + 5T^{2} \)
11 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 + (-0.0309 + 0.0536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.92 - 3.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.09 - 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.75 - 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.755 + 1.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.22 + 7.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.61 + 2.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.95 + 5.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.80 + 4.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.182758337634850353069044822859, −7.83074574224125481603582768945, −7.09491262433252997765057915027, −6.04471261539503732073291885954, −5.24982907480455896767170793983, −4.57167350537345144649726951369, −3.66898634168671207133253080841, −2.67876008326401883077781134292, −1.61220166574468911965902281965, −0.17091861064912731097742151325, 1.51531621847283679693906226924, 2.28876544850719587946546315293, 3.92904536400960711884664631292, 4.07466105765457491681835242621, 5.10932273054964173256011656056, 5.99608119532253908987000611556, 6.95684551240009033533319047436, 7.53885972869489807296000504481, 8.257928620381421478382914959645, 8.874063629709997702774468999634

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