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Properties

Label 2-3e4-9.4-c7-0-16
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  + (3.67 + 6.37i)2-s + (36.9 − 63.9i)4-s + (−234. + 405. i)5-s + (−655. − 1.13e3i)7-s + 1.48e3·8-s − 3.44e3·10-s + (657. + 1.13e3i)11-s + (3.31e3 − 5.73e3i)13-s + (4.82e3 − 8.36e3i)14-s + (740. + 1.28e3i)16-s − 9.81e3·17-s + 5.47e4·19-s + (1.72e4 + 2.99e4i)20-s + (−4.84e3 + 8.38e3i)22-s + (2.70e4 − 4.67e4i)23-s + ⋯
L(s)  = 1  + (0.325 + 0.563i)2-s + (0.288 − 0.499i)4-s + (−0.838 + 1.45i)5-s + (−0.722 − 1.25i)7-s + 1.02·8-s − 1.09·10-s + (0.149 + 0.258i)11-s + (0.418 − 0.724i)13-s + (0.470 − 0.814i)14-s + (0.0451 + 0.0782i)16-s − 0.484·17-s + 1.83·19-s + (0.483 + 0.837i)20-s + (−0.0969 + 0.167i)22-s + (0.462 − 0.801i)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

\(L(4)\) \(\approx\) \(1.95370 - 0.344490i\)
\(L(\frac12)\) \(\approx\) \(1.95370 - 0.344490i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-3.67 - 6.37i)T + (-64 + 110. i)T^{2} \)
5 \( 1 + (234. - 405. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (655. + 1.13e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-657. - 1.13e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-3.31e3 + 5.73e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 9.81e3T + 4.10e8T^{2} \)
19 \( 1 - 5.47e4T + 8.93e8T^{2} \)
23 \( 1 + (-2.70e4 + 4.67e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (4.64e4 + 8.05e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (-1.01e4 + 1.76e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 4.81e5T + 9.49e10T^{2} \)
41 \( 1 + (-1.27e5 + 2.20e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-1.19e5 - 2.07e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (4.04e4 + 7.00e4i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 - 1.88e6T + 1.17e12T^{2} \)
59 \( 1 + (2.48e5 - 4.30e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.45e6 + 2.51e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-4.59e4 + 7.96e4i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 5.17e6T + 9.09e12T^{2} \)
73 \( 1 + 2.40e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.14e5 - 3.71e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (3.60e6 + 6.24e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 7.61e6T + 4.42e13T^{2} \)
97 \( 1 + (2.75e6 + 4.76e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−13.25407051539868635600895283883, −11.48512603755283250863378961709, −10.67734762029883475926983781575, −9.900924750544351519642055391360, −7.59427749240331273921296655380, −7.10265713165922919622558161462, −6.06736357074143469500294984636, −4.21373161546134471349071486538, −2.99159713772927634702172664944, −0.67710842644133177037211526014, 1.26199571499836896148870384866, 2.99754702360734652722141436219, 4.22836098224593314973502236679, 5.56812910081296565261412022896, 7.36867402922030529568660723032, 8.642819059523016828536108802290, 9.381892920912932014295023371336, 11.46116128673710230483064003326, 11.87163072980159935591106229120, 12.77357877799593901941526306934

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