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Properties

Label 2-3e4-9.4-c7-0-21
Degree $2$
Conductor $81$
Sign $0.939 + 0.342i$
Analytic cond. $25.3031$
Root an. cond. $5.03022$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.6 + 18.3i)2-s + (−160. + 278. i)4-s + (176. − 305. i)5-s + (−381. − 660. i)7-s − 4.10e3·8-s + 7.46e3·10-s + (−3.57e3 − 6.19e3i)11-s + (3.56e3 − 6.18e3i)13-s + (8.08e3 − 1.40e4i)14-s + (−2.28e4 − 3.96e4i)16-s − 1.02e4·17-s − 3.17e4·19-s + (5.66e4 + 9.80e4i)20-s + (7.57e4 − 1.31e5i)22-s + (2.46e4 − 4.27e4i)23-s + ⋯
L(s)  = 1  + (0.936 + 1.62i)2-s + (−1.25 + 2.17i)4-s + (0.630 − 1.09i)5-s + (−0.420 − 0.728i)7-s − 2.83·8-s + 2.36·10-s + (−0.809 − 1.40i)11-s + (0.450 − 0.780i)13-s + (0.787 − 1.36i)14-s + (−1.39 − 2.42i)16-s − 0.505·17-s − 1.06·19-s + (1.58 + 2.74i)20-s + (1.51 − 2.62i)22-s + (0.422 − 0.732i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(25.3031\)
Root analytic conductor: \(5.03022\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.73456 - 0.305851i\)
\(L(\frac12)\) \(\approx\) \(1.73456 - 0.305851i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-10.6 - 18.3i)T + (-64 + 110. i)T^{2} \)
5 \( 1 + (-176. + 305. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (381. + 660. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (3.57e3 + 6.19e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-3.56e3 + 6.18e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 1.02e4T + 4.10e8T^{2} \)
19 \( 1 + 3.17e4T + 8.93e8T^{2} \)
23 \( 1 + (-2.46e4 + 4.27e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-3.02e4 - 5.24e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (7.81e4 - 1.35e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + 1.12e5T + 9.49e10T^{2} \)
41 \( 1 + (2.93e5 - 5.07e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (8.09e4 + 1.40e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (6.54e4 + 1.13e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 - 1.63e6T + 1.17e12T^{2} \)
59 \( 1 + (4.02e4 - 6.96e4i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (8.62e5 + 1.49e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.17e6 + 2.02e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 4.69e5T + 9.09e12T^{2} \)
73 \( 1 - 3.76e6T + 1.10e13T^{2} \)
79 \( 1 + (4.15e6 + 7.20e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-4.94e5 - 8.56e5i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 - 9.33e6T + 4.42e13T^{2} \)
97 \( 1 + (4.44e6 + 7.69e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−13.26673593626155332455041793797, −12.61151015048593345710129394943, −10.62226660467434318266010634445, −8.808222649685815643210045610549, −8.232719109284651140574734082058, −6.71940652019998921704948758547, −5.70909951808187547634156746537, −4.79022001890079307007717204005, −3.36483003056155425823022086163, −0.40161657629174200820167759277, 1.99832395048197413184667369418, 2.58207525158986326741353747172, 4.09203874937700816007836137045, 5.50077274336088818169338284082, 6.71563309606019321637763057679, 9.167525546946868444391303609668, 10.09855430825885615174263249166, 10.86143991176091347535440026087, 11.94647144026807954216175116380, 12.92951876315786537838129397591

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