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Properties

Label 2-3e4-9.4-c7-0-18
Degree $2$
Conductor $81$
Sign $0.766 - 0.642i$
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  + (8.29 + 14.3i)2-s + (−73.6 + 127. i)4-s + (57.0 − 98.8i)5-s + (−719. − 1.24e3i)7-s − 320.·8-s + 1.89e3·10-s + (2.96e3 + 5.13e3i)11-s + (5.72e3 − 9.91e3i)13-s + (1.19e4 − 2.06e4i)14-s + (6.76e3 + 1.17e4i)16-s + 2.02e4·17-s − 6.35e3·19-s + (8.41e3 + 1.45e4i)20-s + (−4.91e4 + 8.51e4i)22-s + (3.79e4 − 6.56e4i)23-s + ⋯
L(s)  = 1  + (0.733 + 1.27i)2-s + (−0.575 + 0.996i)4-s + (0.204 − 0.353i)5-s + (−0.792 − 1.37i)7-s − 0.221·8-s + 0.599·10-s + (0.671 + 1.16i)11-s + (0.722 − 1.25i)13-s + (1.16 − 2.01i)14-s + (0.413 + 0.715i)16-s + 0.998·17-s − 0.212·19-s + (0.235 + 0.407i)20-s + (−0.984 + 1.70i)22-s + (0.649 − 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.766 - 0.642i$
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: \((2,\ 81,\ (\ :7/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.86791 + 1.04383i\)
\(L(\frac12)\) \(\approx\) \(2.86791 + 1.04383i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-8.29 - 14.3i)T + (-64 + 110. i)T^{2} \)
5 \( 1 + (-57.0 + 98.8i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (719. + 1.24e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-2.96e3 - 5.13e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-5.72e3 + 9.91e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 - 2.02e4T + 4.10e8T^{2} \)
19 \( 1 + 6.35e3T + 8.93e8T^{2} \)
23 \( 1 + (-3.79e4 + 6.56e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (3.73e4 + 6.47e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (-9.46e4 + 1.63e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + 3.34e4T + 9.49e10T^{2} \)
41 \( 1 + (-7.06e4 + 1.22e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-1.23e5 - 2.13e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-1.67e5 - 2.90e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 1.65e6T + 1.17e12T^{2} \)
59 \( 1 + (-1.02e6 + 1.77e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-2.95e5 - 5.11e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (2.67e4 - 4.63e4i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 4.95e6T + 9.09e12T^{2} \)
73 \( 1 - 8.17e5T + 1.10e13T^{2} \)
79 \( 1 + (3.78e6 + 6.55e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (5.09e5 + 8.82e5i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 1.37e6T + 4.42e13T^{2} \)
97 \( 1 + (-5.30e6 - 9.18e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−13.13505930950806929280070297167, −12.68372106825080849747720957058, −10.69007959094685656574883304753, −9.666911919440458896662098094922, −7.995041841250873261047577050087, −7.05021905278371807614594404258, −6.08264685069164416558688674615, −4.70924464777523734253800048001, −3.60513770763791984470470695147, −0.948979629451854245493220394137, 1.35729216525320073169786058208, 2.80394686992945066770321098049, 3.67675125820217519040409566759, 5.45345918739039282150495541183, 6.54344262097206123906175491819, 8.717442245752585103139702473012, 9.653406022499396712242101462471, 10.99799773756371992190908133912, 11.78971366658769306590611256834, 12.58009097112933608322711711329

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