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Properties

Label 2-3e4-9.2-c6-0-22
Degree $2$
Conductor $81$
Sign $-0.984 - 0.173i$
Analytic cond. $18.6343$
Root an. cond. $4.31675$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−32 − 55.4i)4-s + (143 − 247. i)7-s + (−253 − 438. i)13-s + (−2.04e3 + 3.54e3i)16-s − 1.05e4·19-s + (−7.81e3 + 1.35e4i)25-s − 1.83e4·28-s + (−1.76e4 − 3.05e4i)31-s − 8.92e4·37-s + (−5.56e4 + 9.64e4i)43-s + (1.79e4 + 3.10e4i)49-s + (−1.61e4 + 2.80e4i)52-s + (2.10e5 − 3.64e5i)61-s + 2.62e5·64-s + (−8.64e4 − 1.49e5i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.416 − 0.722i)7-s + (−0.115 − 0.199i)13-s + (−0.499 + 0.866i)16-s − 1.54·19-s + (−0.5 + 0.866i)25-s − 0.833·28-s + (−0.592 − 1.02i)31-s − 1.76·37-s + (−0.700 + 1.21i)43-s + (0.152 + 0.263i)49-s + (−0.115 + 0.199i)52-s + (0.927 − 1.60i)61-s + 0.999·64-s + (−0.287 − 0.497i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(18.6343\)
Root analytic conductor: \(4.31675\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :3),\ -0.984 - 0.173i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0406617 + 0.464766i\)
\(L(\frac12)\) \(\approx\) \(0.0406617 + 0.464766i\)
\(L(4)\) 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 \( 1 + (32 + 55.4i)T^{2} \)
5 \( 1 + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-143 + 247. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (253 + 438. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 2.41e7T^{2} \)
19 \( 1 + 1.05e4T + 4.70e7T^{2} \)
23 \( 1 + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.76e4 + 3.05e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + 8.92e4T + 2.56e9T^{2} \)
41 \( 1 + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (5.56e4 - 9.64e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 2.21e10T^{2} \)
59 \( 1 + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.10e5 + 3.64e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (8.64e4 + 1.49e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 1.28e11T^{2} \)
73 \( 1 - 6.38e5T + 1.51e11T^{2} \)
79 \( 1 + (-1.02e5 + 1.77e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 4.96e11T^{2} \)
97 \( 1 + (-2.82e4 + 4.88e4i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−12.80837273592003777307981380067, −11.20015848646792680251567178276, −10.40875893246579428867332986672, −9.303114437296269617519942390450, −8.038010923848080228951327438028, −6.57019542454504169642404306460, −5.21271796889171959105212281543, −4.00195459327551411917969903251, −1.72810013514798661504621029329, −0.16879197340998936023880176114, 2.23526932620029805615265735536, 3.87556032731495333838559028486, 5.19834768057936481162369018214, 6.85682960341544215736144500300, 8.300452241556915263950583023571, 8.917897484162901537898876398709, 10.43197203104563639867193335447, 11.83955507305472634257739474092, 12.51594357608557916505826408948, 13.64761775743518499023999947156

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