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Properties

Label 2-3e4-9.5-c8-0-28
Degree $2$
Conductor $81$
Sign $-0.342 + 0.939i$
Analytic cond. $32.9976$
Root an. cond. $5.74435$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.4 − 14.6i)2-s + (304 − 526. i)4-s + (712. + 411. i)5-s + (−983.5 − 1.70e3i)7-s − 1.03e4i·8-s + 2.41e4·10-s + (−1.08e4 + 6.29e3i)11-s + (2.27e4 − 3.94e4i)13-s + (−5.00e4 − 2.89e4i)14-s + (−7.42e4 − 1.28e5i)16-s − 5.96e4i·17-s + 1.52e5·19-s + (4.33e5 − 2.50e5i)20-s + (−1.84e5 + 3.20e5i)22-s + (1.13e5 + 6.56e4i)23-s + ⋯
L(s)  = 1  + (1.59 − 0.918i)2-s + (1.18 − 2.05i)4-s + (1.14 + 0.658i)5-s + (−0.409 − 0.709i)7-s − 2.52i·8-s + 2.41·10-s + (−0.744 + 0.429i)11-s + (0.796 − 1.37i)13-s + (−1.30 − 0.752i)14-s + (−1.13 − 1.96i)16-s − 0.713i·17-s + 1.16·19-s + (2.70 − 1.56i)20-s + (−0.789 + 1.36i)22-s + (0.406 + 0.234i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(32.9976\)
Root analytic conductor: \(5.74435\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :4),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.40355 - 4.86078i\)
\(L(\frac12)\) \(\approx\) \(3.40355 - 4.86078i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-25.4 + 14.6i)T + (128 - 221. i)T^{2} \)
5 \( 1 + (-712. - 411. i)T + (1.95e5 + 3.38e5i)T^{2} \)
7 \( 1 + (983.5 + 1.70e3i)T + (-2.88e6 + 4.99e6i)T^{2} \)
11 \( 1 + (1.08e4 - 6.29e3i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + (-2.27e4 + 3.94e4i)T + (-4.07e8 - 7.06e8i)T^{2} \)
17 \( 1 + 5.96e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.52e5T + 1.69e10T^{2} \)
23 \( 1 + (-1.13e5 - 6.56e4i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + (5.09e5 - 2.94e5i)T + (2.50e11 - 4.33e11i)T^{2} \)
31 \( 1 + (-8.21e4 + 1.42e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + 6.63e5T + 3.51e12T^{2} \)
41 \( 1 + (-8.12e5 - 4.69e5i)T + (3.99e12 + 6.91e12i)T^{2} \)
43 \( 1 + (2.87e5 + 4.98e5i)T + (-5.84e12 + 1.01e13i)T^{2} \)
47 \( 1 + (7.99e6 - 4.61e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 - 1.03e7iT - 6.22e13T^{2} \)
59 \( 1 + (-4.36e6 - 2.51e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-9.60e6 - 1.66e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-2.99e5 + 5.17e5i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 2.92e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.28e7T + 8.06e14T^{2} \)
79 \( 1 + (-1.17e7 - 2.04e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (-2.89e7 + 1.67e7i)T + (1.12e15 - 1.95e15i)T^{2} \)
89 \( 1 + 2.82e7iT - 3.93e15T^{2} \)
97 \( 1 + (6.82e7 + 1.18e8i)T + (-3.91e15 + 6.78e15i)T^{2} \)
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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

−12.79050577280421776201627179869, −11.30431461026446242842591964433, −10.41571170682040141510629263326, −9.798896288008967586185600070921, −7.24386836750511443779637780404, −5.95775917723816383732082545175, −5.13146208090468048727092180639, −3.43524444481153067024501430182, −2.61336520083607999854862484340, −1.11637042076213741673761342588, 1.99399241660350366866713959910, 3.47718258096914453659246642738, 5.02203283637672483115106277881, 5.79737468239884109229724063311, 6.66325230025739260424999353779, 8.298272524065041022044410712408, 9.467241754996192991242606249924, 11.35316681000568789343291472602, 12.49481343785249503135280411659, 13.34450120501815180417643526288

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