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Properties

Label 2-3e4-9.4-c3-0-7
Degree $2$
Conductor $81$
Sign $0.173 + 0.984i$
Analytic cond. $4.77915$
Root an. cond. $2.18612$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)4-s + (−10 − 17.3i)7-s + (35 − 60.6i)13-s + (−31.9 − 55.4i)16-s + 56·19-s + (62.5 + 108. i)25-s − 160·28-s + (−154 + 266. i)31-s + 110·37-s + (260 + 450. i)43-s + (−28.5 + 49.3i)49-s + (−279. − 484. i)52-s + (−91 − 157. i)61-s − 511.·64-s + (440 − 762. i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.539 − 0.935i)7-s + (0.746 − 1.29i)13-s + (−0.499 − 0.866i)16-s + 0.676·19-s + (0.5 + 0.866i)25-s − 1.07·28-s + (−0.892 + 1.54i)31-s + 0.488·37-s + (0.922 + 1.59i)43-s + (−0.0830 + 0.143i)49-s + (−0.746 − 1.29i)52-s + (−0.191 − 0.330i)61-s − 0.999·64-s + (0.802 − 1.38i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(4.77915\)
Root analytic conductor: \(2.18612\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16433 - 0.976993i\)
\(L(\frac12)\) \(\approx\) \(1.16433 - 0.976993i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (10 + 17.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-35 + 60.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 56T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 110T + 5.06e4T^{2} \)
41 \( 1 + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-260 - 450. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (91 + 157. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-440 + 762. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 + (442 + 765. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + (-665 - 1.15e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.67205891879296383022914018747, −12.67794744694093269701350429329, −11.09649418113292071157968218377, −10.46486176057442141882785033100, −9.366252592733824240875454567511, −7.66077337217139312735823590071, −6.50962800337661463729753899125, −5.25857134290104703764989991109, −3.29834524140959792533822859918, −1.01074200239301532310808889841, 2.39291743634212816315688388098, 3.93765622861054681763989395686, 5.95553850653132572987946564260, 7.09371932708403201654999255702, 8.494520878155399589511029322017, 9.429885403882027162212892880617, 11.12000857279917580024689731832, 11.96263558353518765454484183939, 12.84701981892329407017990759614, 14.00105800545806592779274477840

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