↑

Properties

Label 2-810-5.4-c1-0-19
Degree $2$
Conductor $810$
Sign $-0.894 - 0.447i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + (−1 + 2i)5-s + i·7-s + i·8-s + (2 + i)10-s − 2·11-s − 6i·13-s + 14-s + 16-s + 2i·17-s − 6·19-s + (1 − 2i)20-s + 2i·22-s + i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s + 0.377i·7-s + 0.353i·8-s + (0.632 + 0.316i)10-s − 0.603·11-s − 1.66i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s − 1.37·19-s + (0.223 − 0.447i)20-s + 0.426i·22-s + 0.208i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (1 - 2i)T \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.17883907488857588694831212536, −8.927895939250911386391102622991, −8.090914453733625316899062029754, −7.38424231783881445130406252641, −6.10521963207707746450126980464, −5.29633122825214424114841254839, −3.95206760983480762547768841836, −3.10427971779357953583827368712, −2.11786822605608749364722969462, 0, 1.86627265350834450046512039765, 3.78168104628012741309522551527, 4.52829153612623436996189583037, 5.35628086966803970879463285667, 6.54201123385877517129664791782, 7.28747656041900917098478454085, 8.193778223586007174045619339017, 8.924112324095231111501790289339, 9.586506805980223172737277751495

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