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Properties

Label 2-3e4-9.4-c1-0-0
Degree $2$
Conductor $81$
Sign $0.342 - 0.939i$
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  + (0.866 + 1.5i)2-s + (−0.5 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (−1 − 1.73i)7-s + 1.73·8-s − 3·10-s + (−1.73 − 3i)11-s + (0.5 − 0.866i)13-s + (1.73 − 3i)14-s + (2.49 + 4.33i)16-s − 5.19·17-s + 2·19-s + (−0.866 − 1.5i)20-s + (3 − 5.19i)22-s + (1.73 − 3i)23-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.250 + 0.433i)4-s + (−0.387 + 0.670i)5-s + (−0.377 − 0.654i)7-s + 0.612·8-s − 0.948·10-s + (−0.522 − 0.904i)11-s + (0.138 − 0.240i)13-s + (0.462 − 0.801i)14-s + (0.624 + 1.08i)16-s − 1.26·17-s + 0.458·19-s + (−0.193 − 0.335i)20-s + (0.639 − 1.10i)22-s + (0.361 − 0.625i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986950 + 0.691069i\)
\(L(\frac12)\) \(\approx\) \(0.986950 + 0.691069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 + 3i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.866 - 1.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (3.46 - 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.46 - 6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-6.92 + 12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.92 + 12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.60070106472155135752845496351, −13.73654198220855231189341045587, −12.92129657343797837868417071107, −11.12161270643292904181564553114, −10.44545798280249175447112710406, −8.559284665351692289005324853885, −7.25675471668951485737925185870, −6.51394804337638602176312893508, −5.07166294607354757338284475118, −3.46636983875570748824133002690, 2.30223289907924439389455355950, 4.03569429808329983165749692300, 5.23834673929669910147529011183, 7.20981090805211652553070547516, 8.710000172223182455253992341055, 9.936034701283110710966056203504, 11.22078562905983776082821707685, 12.12781637590161483901320312142, 12.84465296411444100352201443525, 13.68073750477824096665080456340

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