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Properties

Label 2-810-5.2-c2-0-21
Degree $2$
Conductor $810$
Sign $-0.377 - 0.925i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
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  + (1 + i)2-s + 2i·4-s + (−4.07 + 2.90i)5-s + (7.16 + 7.16i)7-s + (−2 + 2i)8-s + (−6.97 − 1.16i)10-s + 16.2·11-s + (14.7 − 14.7i)13-s + 14.3i·14-s − 4·16-s + (7.49 + 7.49i)17-s + 1.15i·19-s + (−5.80 − 8.14i)20-s + (16.2 + 16.2i)22-s + (−14.0 + 14.0i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.814 + 0.580i)5-s + (1.02 + 1.02i)7-s + (−0.250 + 0.250i)8-s + (−0.697 − 0.116i)10-s + 1.47·11-s + (1.13 − 1.13i)13-s + 1.02i·14-s − 0.250·16-s + (0.441 + 0.441i)17-s + 0.0607i·19-s + (−0.290 − 0.407i)20-s + (0.737 + 0.737i)22-s + (−0.610 + 0.610i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.377 - 0.925i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ -0.377 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.610995132\)
\(L(\frac12)\) \(\approx\) \(2.610995132\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (4.07 - 2.90i)T \)
good7 \( 1 + (-7.16 - 7.16i)T + 49iT^{2} \)
11 \( 1 - 16.2T + 121T^{2} \)
13 \( 1 + (-14.7 + 14.7i)T - 169iT^{2} \)
17 \( 1 + (-7.49 - 7.49i)T + 289iT^{2} \)
19 \( 1 - 1.15iT - 361T^{2} \)
23 \( 1 + (14.0 - 14.0i)T - 529iT^{2} \)
29 \( 1 - 23.0iT - 841T^{2} \)
31 \( 1 + 5.01T + 961T^{2} \)
37 \( 1 + (-33.0 - 33.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 52.4T + 1.68e3T^{2} \)
43 \( 1 + (24.4 - 24.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (11.3 + 11.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (30.1 - 30.1i)T - 2.80e3iT^{2} \)
59 \( 1 + 0.843iT - 3.48e3T^{2} \)
61 \( 1 - 31.7T + 3.72e3T^{2} \)
67 \( 1 + (24.6 + 24.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 1.68T + 5.04e3T^{2} \)
73 \( 1 + (-100. + 100. i)T - 5.32e3iT^{2} \)
79 \( 1 - 10.4iT - 6.24e3T^{2} \)
83 \( 1 + (-16.4 + 16.4i)T - 6.88e3iT^{2} \)
89 \( 1 - 102. iT - 7.92e3T^{2} \)
97 \( 1 + (-16.0 - 16.0i)T + 9.40e3iT^{2} \)
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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

−10.52643023871833612836318530726, −9.212141197776809363615959567583, −8.222380269846163802799347269556, −7.993924513103275532705549839018, −6.69060351323742122677884862921, −5.98542652557233600816022470084, −5.03039563371515612961681196606, −3.87919871523179155177573664119, −3.15747711786209775180818796866, −1.50135146343610791263754317781, 0.863898039159571649230938667520, 1.72872317835994505017731356817, 3.77057295980053869104415411538, 4.07796902815945241734341223388, 4.93372244769325267350359333119, 6.29029198392924437374050273347, 7.15070137426264481588336064532, 8.180637847607709103306211517714, 8.916179290347257681033658448583, 9.852375481041730764803114101885

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