Properties

Label 2-3e4-9.5-c4-0-11
Degree $2$
Conductor $81$
Sign $-0.342 + 0.939i$
Analytic cond. $8.37296$
Root an. cond. $2.89360$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 13.8i)4-s + (−35.5 − 61.4i)7-s + (168.5 − 291. i)13-s + (−127. − 221. i)16-s − 601·19-s + (−312.5 − 541. i)25-s + 1.13e3·28-s + (−97 + 168. i)31-s − 529·37-s + (1.60e3 + 2.78e3i)43-s + (−1.32e3 + 2.28e3i)49-s + (2.69e3 + 4.66e3i)52-s + (−3.59e3 − 6.23e3i)61-s + 4.09e3·64-s + (−1.45e3 + 2.51e3i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.724 − 1.25i)7-s + (0.997 − 1.72i)13-s + (−0.499 − 0.866i)16-s − 1.66·19-s + (−0.5 − 0.866i)25-s + 1.44·28-s + (−0.100 + 0.174i)31-s − 0.386·37-s + (0.869 + 1.50i)43-s + (−0.549 + 0.952i)49-s + (0.997 + 1.72i)52-s + (−0.967 − 1.67i)61-s + 0.999·64-s + (−0.323 + 0.560i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(8.37296\)
Root analytic conductor: \(2.89360\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :2),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.432726 - 0.617998i\)
\(L(\frac12)\) \(\approx\) \(0.432726 - 0.617998i\)
\(L(3)\) 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 \( 1 + (8 - 13.8i)T^{2} \)
5 \( 1 + (312.5 + 541. i)T^{2} \)
7 \( 1 + (35.5 + 61.4i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-168.5 + 291. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 8.35e4T^{2} \)
19 \( 1 + 601T + 1.30e5T^{2} \)
23 \( 1 + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (97 - 168. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 529T + 1.87e6T^{2} \)
41 \( 1 + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.60e3 - 2.78e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 7.89e6T^{2} \)
59 \( 1 + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (3.59e3 + 6.23e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.45e3 - 2.51e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.54e7T^{2} \)
73 \( 1 + 1.24e3T + 2.83e7T^{2} \)
79 \( 1 + (2.33e3 + 4.05e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 6.27e7T^{2} \)
97 \( 1 + (4.53e3 + 7.85e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.09516018904253517801615138367, −12.64983328405788794463869124049, −10.93720707151669629777714345889, −10.05100088573048370879608789769, −8.563035203194021190923595709796, −7.64669056413070979617927312834, −6.25103316880298593900780465613, −4.30829255519721293153534732726, −3.22892814327337813813069071862, −0.37490054776353981793614398891, 1.95008840979031324587869547035, 4.12205400580966726825238146755, 5.73035025046356167552515514518, 6.57629322554324028343004282048, 8.807667928680595782478912815883, 9.212114724144471015767174281082, 10.60323299408741565936592400739, 11.76850859752348433369143610723, 12.98736411383849658028550935122, 13.94579870095200770565710636896

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