Properties

Label 2-12e3-9.2-c2-0-39
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: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.450 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.379772284\)
\(L(\frac12)\) \(\approx\) \(1.379772284\)
\(L(2)\) 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

−9.215460780523069730175228556850, −8.209450975200745045940198740154, −7.32495601682502212507963001801, −6.10473661985494391382441382310, −5.64425779641710426578729046164, −5.14507533088394652117015508120, −3.75386361707268597468417456784, −2.54180178686418006063698110219, −1.89068879137958828262916447284, −0.33563588511162485271406719555, 1.38622231767998278882000173551, 2.38863506490172296380219124004, 3.43450793690496794669187341385, 4.32215330470273092374758945743, 5.54779004165535702145141752503, 6.23102496539151679262892768679, 7.02384794380299004660124293967, 7.46447004660598231701890093220, 8.845041663437610704269003975673, 9.541001256113953555176524361740

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