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Properties

Label 2-3e4-9.7-c7-0-15
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: $\chi_{81} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.66348 - 0.969427i\)
\(L(\frac12)\) \(\approx\) \(2.66348 - 0.969427i\)
\(L(\frac{9}{2})\) 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

−12.61917364408131415978440728980, −11.91947191266443168026991138322, −10.69419956343712866905906664714, −9.814955851637980433155283744899, −7.83630361185095359034289856860, −7.42057780900173510192859657890, −5.28146039116548533953762750929, −4.03251764576108916159645197968, −2.68322537880727914996201282083, −1.07639652773887397061009406295, 1.21805815557966426991407645997, 2.97376639608055154093305193977, 4.98908821232991632786516112457, 5.81153112250028029100647198888, 7.14647858863026412466020135547, 8.274948099258722391967896559118, 9.757230966500861880099895783873, 10.94621338913166233351843407625, 11.87859364193392952338172295162, 13.27671200773275606574726011983

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