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Properties

Label 2-12e3-9.5-c2-0-17
Degree $2$
Conductor $1728$
Sign $-0.450 - 0.892i$
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  + (6.55 + 3.78i)5-s + (4.55 + 7.89i)7-s + (0.383 − 0.221i)11-s + (−5.55 + 9.62i)13-s + 8.01i·17-s − 8.11·19-s + (20.4 + 11.8i)23-s + (16.1 + 28.0i)25-s + (−45.9 + 26.5i)29-s + (14.6 − 25.4i)31-s + 69.0i·35-s − 18.4·37-s + (38.9 + 22.4i)41-s + (−11.5 − 19.9i)43-s + (−7.32 + 4.22i)47-s + ⋯
L(s)  = 1  + (1.31 + 0.757i)5-s + (0.651 + 1.12i)7-s + (0.0348 − 0.0201i)11-s + (−0.427 + 0.740i)13-s + 0.471i·17-s − 0.427·19-s + (0.888 + 0.513i)23-s + (0.647 + 1.12i)25-s + (−1.58 + 0.913i)29-s + (0.473 − 0.819i)31-s + 1.97i·35-s − 0.499·37-s + (0.950 + 0.548i)41-s + (−0.267 − 0.463i)43-s + (−0.155 + 0.0899i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.450 - 0.892i$
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: \((2,\ 1728,\ (\ :1),\ -0.450 - 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.496122845\)
\(L(\frac12)\) \(\approx\) \(2.496122845\)
\(L(2)\) 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 \( 1 + (-6.55 - 3.78i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.55 - 7.89i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.383 + 0.221i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.55 - 9.62i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8.01iT - 289T^{2} \)
19 \( 1 + 8.11T + 361T^{2} \)
23 \( 1 + (-20.4 - 11.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (45.9 - 26.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.6 + 25.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 18.4T + 1.36e3T^{2} \)
41 \( 1 + (-38.9 - 22.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7.32 - 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 60.5iT - 2.80e3T^{2} \)
59 \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.67 - 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.8 + 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35T + 5.32e3T^{2} \)
79 \( 1 + (-0.792 - 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-7.32 + 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (57.6 + 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.279414903450706812390415782787, −8.902312313326118863927084147461, −7.77026740516466395341222725268, −6.89217718700989468776883393828, −6.06096609328125110905433337367, −5.51766409224355456853258871825, −4.61789522192506265873442349219, −3.23543307543339127449363701038, −2.21723285557589791560013993877, −1.69621085995870241671908005841, 0.62117889195143717166227568146, 1.56972630692372080635333849633, 2.61966858589074319941609907190, 3.98779800034119608379362078032, 4.92595840417199134099648864758, 5.43341755359896261437692092492, 6.45513250531712125065766771060, 7.33212121494425640858512391049, 8.093804015580827455395057571019, 8.995730080924955019656394316578

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