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Properties

Label 2-3e4-9.7-c7-0-22
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  + (5.19 − 9i)2-s + (10 + 17.3i)4-s + (−176. − 306i)5-s + (279.5 − 484. i)7-s + 1.53e3·8-s − 3.67e3·10-s + (2.35e3 − 4.08e3i)11-s + (4.33e3 + 7.50e3i)13-s + (−2.90e3 − 5.03e3i)14-s + (6.71e3 − 1.16e4i)16-s − 2.51e4·17-s − 3.24e4·19-s + (3.53e3 − 6.11e3i)20-s + (−2.45e4 − 4.24e4i)22-s + (−4.12e4 − 7.13e4i)23-s + ⋯
L(s)  = 1  + (0.459 − 0.795i)2-s + (0.0781 + 0.135i)4-s + (−0.632 − 1.09i)5-s + (0.307 − 0.533i)7-s + 1.06·8-s − 1.16·10-s + (0.534 − 0.925i)11-s + (0.547 + 0.947i)13-s + (−0.282 − 0.490i)14-s + (0.409 − 0.709i)16-s − 1.24·17-s − 1.08·19-s + (0.0987 − 0.171i)20-s + (−0.490 − 0.850i)22-s + (−0.706 − 1.22i)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: $\chi_{81} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.355769 - 2.01767i\)
\(L(\frac12)\) \(\approx\) \(0.355769 - 2.01767i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-5.19 + 9i)T + (-64 - 110. i)T^{2} \)
5 \( 1 + (176. + 306i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-279.5 + 484. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-2.35e3 + 4.08e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-4.33e3 - 7.50e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + 2.51e4T + 4.10e8T^{2} \)
19 \( 1 + 3.24e4T + 8.93e8T^{2} \)
23 \( 1 + (4.12e4 + 7.13e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-7.88e4 + 1.36e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (1.14e5 + 1.99e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 5.41e5T + 9.49e10T^{2} \)
41 \( 1 + (-1.76e5 - 3.06e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (-2.32e5 + 4.02e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (4.15e5 - 7.19e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 - 1.02e6T + 1.17e12T^{2} \)
59 \( 1 + (3.92e5 + 6.80e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-6.88e4 + 1.19e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.57e5 - 2.71e5i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.80e6T + 9.09e12T^{2} \)
73 \( 1 - 2.66e6T + 1.10e13T^{2} \)
79 \( 1 + (5.50e5 - 9.54e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.03e6 + 5.26e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 - 3.28e6T + 4.42e13T^{2} \)
97 \( 1 + (-1.48e6 + 2.58e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−12.35412867347671198083274646572, −11.47280250174924872128671371992, −10.72453986806688293158787313058, −8.897092064082042586569283525489, −8.094666481790539354377175136529, −6.50375985396780860826174541370, −4.42826214899664373649937632833, −4.00314312759335155632149631482, −2.02546119988824127141718557448, −0.57022887522433816093147511514, 1.92733741506963746283395496947, 3.75021902064097791309476940041, 5.20289791449093354253297837794, 6.55684385941575001368145094602, 7.27118321402318010562454204617, 8.611976445317705237727675699992, 10.39099062078535653818933280622, 11.10096691984898593782534550503, 12.40000024891570573942909518420, 13.74703441614770975753269981279

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