Properties

Label 2-3e4-27.13-c1-0-0
Degree $2$
Conductor $81$
Sign $-0.490 - 0.871i$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.417 + 2.36i)2-s + (−3.54 + 1.28i)4-s + (−0.0713 − 0.0598i)5-s + (0.544 + 0.198i)7-s + (−2.12 − 3.67i)8-s + (0.111 − 0.193i)10-s + (2.36 − 1.98i)11-s + (0.729 − 4.13i)13-s + (−0.241 + 1.37i)14-s + (2.03 − 1.71i)16-s + (−0.995 + 1.72i)17-s + (1.92 + 3.33i)19-s + (0.329 + 0.119i)20-s + (5.69 + 4.77i)22-s + (−4.18 + 1.52i)23-s + ⋯
L(s)  = 1  + (0.294 + 1.67i)2-s + (−1.77 + 0.644i)4-s + (−0.0318 − 0.0267i)5-s + (0.205 + 0.0749i)7-s + (−0.750 − 1.29i)8-s + (0.0353 − 0.0612i)10-s + (0.714 − 0.599i)11-s + (0.202 − 1.14i)13-s + (−0.0646 + 0.366i)14-s + (0.509 − 0.427i)16-s + (−0.241 + 0.418i)17-s + (0.441 + 0.764i)19-s + (0.0736 + 0.0268i)20-s + (1.21 + 1.01i)22-s + (−0.872 + 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.490 - 0.871i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1/2),\ -0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522028 + 0.893362i\)
\(L(\frac12)\) \(\approx\) \(0.522028 + 0.893362i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.417 - 2.36i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.0713 + 0.0598i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.544 - 0.198i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.36 + 1.98i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.729 + 4.13i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.995 - 1.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.18 - 1.52i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.11 + 6.30i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.55 - 0.566i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.01 - 3.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.190 + 1.07i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (5.28 - 4.43i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.37 - 1.22i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 + (-7.87 - 6.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-12.4 - 4.51i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.53 - 8.70i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.572 + 0.991i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0977 + 0.169i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.25 + 7.09i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.58 - 14.6i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.776 + 1.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.05 + 3.40i)T + (16.8 - 95.5i)T^{2} \)
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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

−14.80888014054293749977325951973, −13.98941219242717048836570358849, −13.04518032501323883228659768794, −11.67959131168413229420835973769, −10.02884783058399261150168077370, −8.526836418353905960311181751188, −7.84801545511999548149200677099, −6.38836110722858213220528364097, −5.52058241346123671325103760335, −3.93545098214888768680473724978, 1.86540728154967913402049276079, 3.69122795591946895028701248533, 4.89889003732022284234093291981, 6.97232819643360560934389184074, 8.964226241834396725485239788233, 9.734145238010580559322598672326, 11.06786410921081926809085955749, 11.70385704205471109853982682203, 12.68605482314522015223801863168, 13.78039599186287460948652984720

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