Properties

Label 2-3e4-27.13-c1-0-1
Degree $2$
Conductor $81$
Sign $0.605 + 0.795i$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.183 − 1.03i)2-s + (0.834 − 0.303i)4-s + (1.33 + 1.12i)5-s + (−2.31 − 0.841i)7-s + (−1.52 − 2.63i)8-s + (0.920 − 1.59i)10-s + (0.960 − 0.806i)11-s + (−0.789 + 4.47i)13-s + (−0.450 + 2.55i)14-s + (−1.09 + 0.921i)16-s + (−3.32 + 5.75i)17-s + (−0.124 − 0.215i)19-s + (1.45 + 0.530i)20-s + (−1.01 − 0.849i)22-s + (0.791 − 0.287i)23-s + ⋯
L(s)  = 1  + (−0.129 − 0.734i)2-s + (0.417 − 0.151i)4-s + (0.598 + 0.501i)5-s + (−0.873 − 0.317i)7-s + (−0.538 − 0.932i)8-s + (0.291 − 0.504i)10-s + (0.289 − 0.243i)11-s + (−0.219 + 1.24i)13-s + (−0.120 + 0.682i)14-s + (−0.274 + 0.230i)16-s + (−0.806 + 1.39i)17-s + (−0.0285 − 0.0495i)19-s + (0.325 + 0.118i)20-s + (−0.215 − 0.181i)22-s + (0.164 − 0.0600i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893377 - 0.442838i\)
\(L(\frac12)\) \(\approx\) \(0.893377 - 0.442838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.183 + 1.03i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.33 - 1.12i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.31 + 0.841i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.960 + 0.806i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.789 - 4.47i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.32 - 5.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.124 + 0.215i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.791 + 0.287i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.0889 - 0.504i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.770 + 0.280i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (1.30 - 2.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.41 + 8.02i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.31 + 2.78i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.98 + 1.81i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (-2.30 - 1.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.70 + 0.986i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.75 + 9.93i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.0447 + 0.0774i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.66 - 4.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.829 - 4.70i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.39 + 7.91i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-3.35 - 5.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.20 + 3.52i)T + (16.8 - 95.5i)T^{2} \)
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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

−14.08129204109670002615835982899, −13.01480996156960198889011979614, −11.92977543063548179942743999945, −10.79623880000335488468427631511, −10.04169862870961825985496177880, −8.978463299224768105602215963313, −6.86109191877395207907403819444, −6.22642883041752329690605616524, −3.80003426340643955048360263003, −2.16068297297437154876442698701, 2.78558948457901225601709292024, 5.23343862859717802680960239048, 6.33634652812503533449301928274, 7.48471077020524467910435200652, 8.866165455814851765675026646893, 9.821708437527237203557539705835, 11.34582358952350661749352936068, 12.53518288694509463104389523787, 13.41699920201409352569483764496, 14.79165695925655715215979779928

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