↑

Properties

Label 2-110-5.4-c1-0-3
Degree $2$
Conductor $110$
Sign $0.894 + 0.447i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 2i·3-s − 4-s + (1 − 2i)5-s + 2·6-s − i·8-s − 9-s + (2 + i)10-s + 11-s + 2i·12-s + 2i·13-s + (−4 − 2i)15-s + 16-s + 6i·17-s − i·18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.816·6-s − 0.353i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s + 0.301·11-s + 0.577i·12-s + 0.554i·13-s + (−1.03 − 0.516i)15-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02514 - 0.242004i\)
\(L(\frac12)\) \(\approx\) \(1.02514 - 0.242004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-1 + 2i)T \)
11 \( 1 - T \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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

−13.47034309321741386944171225397, −12.78712411549972235679397588636, −11.99013561235859855557904339369, −10.24458181677422170414821182042, −8.875434907911310490370454252904, −8.108784035382912985059531306216, −6.78836598716420932076224026642, −5.94549742079356113347458994976, −4.38005348521344230562326433019, −1.61079454753014370055305275360, 2.72240186290776383340150330989, 4.06457771373842461193433223764, 5.40861580929093424452721009393, 7.03345679365132223942524460625, 8.818067488690250922978154348263, 9.823303775794516894307125125286, 10.51368487830850423797003666828, 11.30097716749565062820308375457, 12.61410960793302619396857411523, 13.89070008196443244921968353817

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