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Properties

Label 2-3e4-9.4-c7-0-5
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  + (−3.79 − 6.57i)2-s + (35.1 − 60.9i)4-s + (32.9 − 56.9i)5-s + (369. + 639. i)7-s − 1.50e3·8-s − 499.·10-s + (2.48e3 + 4.29e3i)11-s + (−2.98e3 + 5.16e3i)13-s + (2.80e3 − 4.85e3i)14-s + (1.21e3 + 2.10e3i)16-s − 3.66e4·17-s + 2.23e4·19-s + (−2.31e3 − 4.00e3i)20-s + (1.88e4 − 3.26e4i)22-s + (−2.57e4 + 4.45e4i)23-s + ⋯
L(s)  = 1  + (−0.335 − 0.581i)2-s + (0.274 − 0.475i)4-s + (0.117 − 0.203i)5-s + (0.406 + 0.704i)7-s − 1.03·8-s − 0.158·10-s + (0.561 + 0.973i)11-s + (−0.376 + 0.652i)13-s + (0.273 − 0.472i)14-s + (0.0742 + 0.128i)16-s − 1.80·17-s + 0.748·19-s + (−0.0646 − 0.112i)20-s + (0.377 − 0.653i)22-s + (−0.441 + 0.763i)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

\(L(4)\) \(\approx\) \(1.20558 + 0.438798i\)
\(L(\frac12)\) \(\approx\) \(1.20558 + 0.438798i\)
\(L(\frac{9}{2})\) 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 \( 1 + (3.79 + 6.57i)T + (-64 + 110. i)T^{2} \)
5 \( 1 + (-32.9 + 56.9i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-369. - 639. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-2.48e3 - 4.29e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (2.98e3 - 5.16e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 3.66e4T + 4.10e8T^{2} \)
19 \( 1 - 2.23e4T + 8.93e8T^{2} \)
23 \( 1 + (2.57e4 - 4.45e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (3.42e4 + 5.93e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (7.53e4 - 1.30e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 4.89e5T + 9.49e10T^{2} \)
41 \( 1 + (-2.95e5 + 5.11e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-4.21e5 - 7.29e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-6.13e5 - 1.06e6i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 9.58e5T + 1.17e12T^{2} \)
59 \( 1 + (1.58e5 - 2.73e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.48e4 - 2.57e4i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.46e5 - 2.53e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 7.14e5T + 9.09e12T^{2} \)
73 \( 1 + 3.96e6T + 1.10e13T^{2} \)
79 \( 1 + (1.26e6 + 2.19e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-8.31e5 - 1.44e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 4.64e6T + 4.42e13T^{2} \)
97 \( 1 + (7.35e6 + 1.27e7i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−12.75726244421185054636291333618, −11.72032515424199548975913977890, −11.03289941595770332262026320567, −9.501421087559634505642325944026, −9.085976549780191183495623436939, −7.23726401240407476905593892751, −5.93035627402637886080380259787, −4.53213403965099846041649382477, −2.44337210471920643119098780893, −1.45160080976710607710921842289, 0.47277622131960460452056500387, 2.62233453003962693047894731134, 4.15626497689373217558523284511, 6.02199072790136231819718268545, 7.07389507520542995188969958167, 8.130469170762493019243330713547, 9.147000163318842966502704820346, 10.71687778891284670132152218826, 11.56999984533582764836686790519, 12.87878789771925781381537447782

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