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Properties

Label 2-12e3-36.7-c2-0-16
Degree $2$
Conductor $1728$
Sign $0.416 - 0.909i$
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  + (0.355 + 0.615i)5-s + (2.70 + 1.56i)7-s + (14.3 + 8.30i)11-s + (9.17 + 15.8i)13-s + 9.69·17-s − 8.20i·19-s + (−1.94 + 1.12i)23-s + (12.2 − 21.2i)25-s + (−20.8 + 36.0i)29-s + (−21.6 + 12.4i)31-s + 2.21i·35-s + 40.3·37-s + (−25.6 − 44.5i)41-s + (56.6 + 32.7i)43-s + (−29.2 − 16.9i)47-s + ⋯
L(s)  = 1  + (0.0710 + 0.123i)5-s + (0.386 + 0.223i)7-s + (1.30 + 0.754i)11-s + (0.705 + 1.22i)13-s + 0.570·17-s − 0.431i·19-s + (−0.0847 + 0.0489i)23-s + (0.489 − 0.848i)25-s + (−0.717 + 1.24i)29-s + (−0.697 + 0.402i)31-s + 0.0634i·35-s + 1.09·37-s + (−0.626 − 1.08i)41-s + (1.31 + 0.760i)43-s + (−0.623 − 0.359i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.398492004\)
\(L(\frac12)\) \(\approx\) \(2.398492004\)
\(L(2)\) 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 \( 1 + (-0.355 - 0.615i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.70 - 1.56i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-14.3 - 8.30i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.17 - 15.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 9.69T + 289T^{2} \)
19 \( 1 + 8.20iT - 361T^{2} \)
23 \( 1 + (1.94 - 1.12i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (20.8 - 36.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (21.6 - 12.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 40.3T + 1.36e3T^{2} \)
41 \( 1 + (25.6 + 44.5i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-56.6 - 32.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (29.2 + 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 90.6T + 2.80e3T^{2} \)
59 \( 1 + (66.2 - 38.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-34.5 + 19.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 38.1T + 5.32e3T^{2} \)
79 \( 1 + (-94.4 - 54.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-113. - 65.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 38.0T + 7.92e3T^{2} \)
97 \( 1 + (12.1 - 21.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.187912247290921196146067151674, −8.708977132250572176342419128427, −7.62332705714044375955953675638, −6.79437970320270425121494189851, −6.25989230874622105909053182012, −5.10646947249291103740610016022, −4.28019731622473646558569970496, −3.46483619865865914607273146993, −2.07098046591698700813434348143, −1.25207095466556157835306417846, 0.71547188191975716822892170898, 1.62696637143667994111095450337, 3.18631868244848556563572557528, 3.80799210012642552240242923712, 4.89833504842996737242601201453, 5.94163368225720948358314525911, 6.31843321692491130232869199499, 7.77274824658669251784730818056, 7.953076360038412519740042323086, 9.131076130881212329515154672038

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