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Properties

Label 2-3e4-9.7-c7-0-7
Degree $2$
Conductor $81$
Sign $0.173 - 0.984i$
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 + 5.19i)2-s + (46 + 79.6i)4-s + (−195 − 337. i)5-s + (32 − 55.4i)7-s − 1.32e3·8-s + 2.34e3·10-s + (474 − 820. i)11-s + (2.54e3 + 4.41e3i)13-s + (192 + 332. i)14-s + (−1.92e3 + 3.33e3i)16-s + 2.83e4·17-s − 8.62e3·19-s + (1.79e4 − 3.10e4i)20-s + (2.84e3 + 4.92e3i)22-s + (7.64e3 + 1.32e4i)23-s + ⋯
L(s)  = 1  + (−0.265 + 0.459i)2-s + (0.359 + 0.622i)4-s + (−0.697 − 1.20i)5-s + (0.0352 − 0.0610i)7-s − 0.911·8-s + 0.739·10-s + (0.107 − 0.185i)11-s + (0.321 + 0.557i)13-s + (0.0187 + 0.0323i)14-s + (−0.117 + 0.203i)16-s + 1.40·17-s − 0.288·19-s + (0.501 − 0.868i)20-s + (0.0569 + 0.0986i)22-s + (0.131 + 0.226i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(25.3031\)
Root analytic conductor: \(5.03022\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.08981 + 0.914465i\)
\(L(\frac12)\) \(\approx\) \(1.08981 + 0.914465i\)
\(L(\frac{9}{2})\) 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 \( 1 + (3 - 5.19i)T + (-64 - 110. i)T^{2} \)
5 \( 1 + (195 + 337. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-32 + 55.4i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-474 + 820. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-2.54e3 - 4.41e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 2.83e4T + 4.10e8T^{2} \)
19 \( 1 + 8.62e3T + 8.93e8T^{2} \)
23 \( 1 + (-7.64e3 - 1.32e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (1.82e4 - 3.16e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (-1.38e5 - 2.39e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 2.68e5T + 9.49e10T^{2} \)
41 \( 1 + (-3.14e5 - 5.45e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (3.42e5 - 5.93e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (2.91e5 - 5.05e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 4.28e5T + 1.17e12T^{2} \)
59 \( 1 + (6.53e5 + 1.13e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.50e5 - 2.60e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-2.53e5 - 4.39e5i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 5.56e6T + 9.09e12T^{2} \)
73 \( 1 - 1.36e6T + 1.10e13T^{2} \)
79 \( 1 + (-3.45e6 + 5.98e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-2.18e6 + 3.79e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + 8.52e6T + 4.42e13T^{2} \)
97 \( 1 + (-4.41e6 + 7.64e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−12.86902761873667888823174092314, −12.17737686359930592672743205534, −11.25888760112186086294701085431, −9.457432660188562358321096651550, −8.400121828715746540671205924802, −7.70926737391755823941387744782, −6.26912618919320293550261903277, −4.67396488317528629529747915858, −3.29656150328313478376054519022, −1.11210981578626505466538723934, 0.62040112999915299392346281810, 2.40589340426000954177657622317, 3.61870046068402110000549872473, 5.67727075398821802535828907984, 6.85842242960574059753073927991, 8.040272922280292031462158884487, 9.712181534914462576202066887373, 10.57237691272699703989176721023, 11.38655811870446620277468566904, 12.30825338308689239542730527839

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