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Properties

Label 2-12e3-9.5-c2-0-41
Degree $2$
Conductor $1728$
Sign $-0.893 + 0.449i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 + 1.04i)5-s + (0.781 + 1.35i)7-s + (10.8 − 6.25i)11-s + (−11.0 + 19.1i)13-s − 12.6i·17-s − 21.7·19-s + (−28.7 − 16.6i)23-s + (−10.3 − 17.8i)25-s + (−25.7 + 14.8i)29-s + (−6.91 + 11.9i)31-s + 3.26i·35-s + 8.26·37-s + (−43.8 − 25.3i)41-s + (35.5 + 61.5i)43-s + (57.2 − 33.0i)47-s + ⋯
L(s)  = 1  + (0.361 + 0.208i)5-s + (0.111 + 0.193i)7-s + (0.984 − 0.568i)11-s + (−0.849 + 1.47i)13-s − 0.747i·17-s − 1.14·19-s + (−1.25 − 0.722i)23-s + (−0.412 − 0.714i)25-s + (−0.888 + 0.513i)29-s + (−0.223 + 0.386i)31-s + 0.0932i·35-s + 0.223·37-s + (−1.06 − 0.617i)41-s + (0.826 + 1.43i)43-s + (1.21 − 0.703i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.893 + 0.449i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2777914167\)
\(L(\frac12)\) \(\approx\) \(0.2777914167\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-1.80 - 1.04i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.781 - 1.35i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.8 + 6.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (11.0 - 19.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 12.6iT - 289T^{2} \)
19 \( 1 + 21.7T + 361T^{2} \)
23 \( 1 + (28.7 + 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (25.7 - 14.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (6.91 - 11.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 8.26T + 1.36e3T^{2} \)
41 \( 1 + (43.8 + 25.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 - 61.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-57.2 + 33.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 6.04iT - 2.80e3T^{2} \)
59 \( 1 + (-8.01 - 4.62i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (51.9 + 89.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (19.8 - 34.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 18.3iT - 5.04e3T^{2} \)
73 \( 1 + 68.5T + 5.32e3T^{2} \)
79 \( 1 + (13.3 + 23.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (21.0 - 12.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + (2.51 + 4.35i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.958712691218918393484094806119, −8.101856607126691896472532019711, −6.98293483625265881475463090189, −6.50238516979289722427285052974, −5.65582598956262590346189255331, −4.51528191646484364793063894816, −3.88563288466747547451197551898, −2.49389780739279722177409394365, −1.74983691617457404528609120518, −0.06757730456858675625325337391, 1.45245661409859772613231178396, 2.38646097009609461877663783173, 3.74273968079015240718389142484, 4.39990489422796241084192224727, 5.62656212726964026132905641401, 6.03532919237455384877710214358, 7.27881431966482640271199180915, 7.77071542708473347490993104721, 8.746377212691884574262925811869, 9.521603084532622356579166133350

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