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Properties

Label 2-3e4-9.5-c2-0-4
Degree $2$
Conductor $81$
Sign $0.642 + 0.766i$
Analytic cond. $2.20709$
Root an. cond. $1.48562$
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  + (2.59 − 1.5i)2-s + (2.5 − 4.33i)4-s + (2.59 + 1.5i)5-s + (−2.5 − 4.33i)7-s − 3.00i·8-s + 9·10-s + (−12.9 + 7.5i)11-s + (5 − 8.66i)13-s + (−12.9 − 7.50i)14-s + (5.49 + 9.52i)16-s + 18i·17-s − 16·19-s + (12.9 − 7.50i)20-s + (−22.5 + 38.9i)22-s + (10.3 + 6i)23-s + ⋯
L(s)  = 1  + (1.29 − 0.750i)2-s + (0.625 − 1.08i)4-s + (0.519 + 0.300i)5-s + (−0.357 − 0.618i)7-s − 0.375i·8-s + 0.900·10-s + (−1.18 + 0.681i)11-s + (0.384 − 0.666i)13-s + (−0.927 − 0.535i)14-s + (0.343 + 0.595i)16-s + 1.05i·17-s − 0.842·19-s + (0.649 − 0.375i)20-s + (−1.02 + 1.77i)22-s + (0.451 + 0.260i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(2.20709\)
Root analytic conductor: \(1.48562\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.11872 - 0.987975i\)
\(L(\frac12)\) \(\approx\) \(2.11872 - 0.987975i\)
\(L(2)\) 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 \( 1 + (-2.59 + 1.5i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-2.59 - 1.5i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.5 + 4.33i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.9 - 7.5i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5 + 8.66i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 18iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 + (-10.3 - 6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-25.9 + 15i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 20T + 1.36e3T^{2} \)
41 \( 1 + (51.9 + 30i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (25 + 43.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.19 - 3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 27iT - 2.80e3T^{2} \)
59 \( 1 + (-25.9 - 15i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-38 - 65.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 90iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-2.59 + 1.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 90iT - 7.92e3T^{2} \)
97 \( 1 + (-42.5 - 73.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.50819510011103990299571006322, −13.13614627240444823249364886647, −12.10241969378967895787446763042, −10.55197058322098728258570655403, −10.29336940848671096953234921168, −8.176168010470361783117948095417, −6.47671806090533752953048951969, −5.23133751067933461261997075393, −3.82085812329847681431098461583, −2.33498666579487863184302867623, 2.95475049556169323779985259683, 4.78705228841543320080764017958, 5.75916559617087562587142684433, 6.80871133424025378656369001083, 8.397055907520961843999267366502, 9.736051030952389704757090299917, 11.32700713719374946404044604746, 12.63104411957248250239142257234, 13.33130561374307585942883968821, 14.10971730939407798609093495824

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