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Properties

Label 2-3e4-9.5-c2-0-1
Degree $2$
Conductor $81$
Sign $0.342 - 0.939i$
Analytic cond. $2.20709$
Root an. cond. $1.48562$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)4-s + (6.5 + 11.2i)7-s + (0.5 − 0.866i)13-s + (−7.99 − 13.8i)16-s + 11·19-s + (−12.5 − 21.6i)25-s − 51.9·28-s + (23 − 39.8i)31-s + 47·37-s + (11 + 19.0i)43-s + (−59.9 + 103. i)49-s + (1.99 + 3.46i)52-s + (60.5 + 104. i)61-s + 63.9·64-s + (54.5 − 94.3i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.928 + 1.60i)7-s + (0.0384 − 0.0666i)13-s + (−0.499 − 0.866i)16-s + 0.578·19-s + (−0.5 − 0.866i)25-s − 1.85·28-s + (0.741 − 1.28i)31-s + 1.27·37-s + (0.255 + 0.443i)43-s + (−1.22 + 2.12i)49-s + (0.0384 + 0.0666i)52-s + (0.991 + 1.71i)61-s + 0.999·64-s + (0.813 − 1.40i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(2.20709\)
Root analytic conductor: \(1.48562\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.978997 + 0.685501i\)
\(L(\frac12)\) \(\approx\) \(0.978997 + 0.685501i\)
\(L(2)\) 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 \( 1 + (2 - 3.46i)T^{2} \)
5 \( 1 + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.5 - 11.2i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23 + 39.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 47T + 1.36e3T^{2} \)
41 \( 1 + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-11 - 19.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-60.5 - 104. i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.5 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 + (65.5 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (83.5 + 144. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.33040299481517925741382520613, −13.14702849966106925948228184211, −12.05520007334758069228402019447, −11.47190224230496509220850200051, −9.604973070982991620814282543154, −8.566984703891580629945732962698, −7.76969061293628736998074105445, −5.83590830472550467809345406483, −4.50190154096830105490149964740, −2.59908910368367246631669171122, 1.18165756633759695799175678044, 4.09380848723124805829208855825, 5.24819206200477020327456975966, 6.91970837959825693505693859302, 8.157560518323420213261202940739, 9.634267482006436341292137157758, 10.57531606285309577624132716816, 11.43586352432668590115109270239, 13.19188169398870890671628921819, 14.00233333924651024747736730076

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