↑

Properties

Label 2-55-5.2-c2-0-8
Degree $2$
Conductor $55$
Sign $-0.967 + 0.253i$
Analytic cond. $1.49864$
Root an. cond. $1.22419$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.864 − 0.864i)2-s + (−2.45 + 2.45i)3-s − 2.50i·4-s + (−4.77 − 1.46i)5-s + 4.24·6-s + (−6.72 − 6.72i)7-s + (−5.62 + 5.62i)8-s − 3.03i·9-s + (2.86 + 5.40i)10-s + 3.31·11-s + (6.14 + 6.14i)12-s + (0.519 − 0.519i)13-s + 11.6i·14-s + (15.3 − 8.12i)15-s − 0.288·16-s + (5.04 + 5.04i)17-s + ⋯
L(s)  = 1  + (−0.432 − 0.432i)2-s + (−0.817 + 0.817i)3-s − 0.626i·4-s + (−0.955 − 0.293i)5-s + 0.707·6-s + (−0.960 − 0.960i)7-s + (−0.703 + 0.703i)8-s − 0.337i·9-s + (0.286 + 0.540i)10-s + 0.301·11-s + (0.511 + 0.511i)12-s + (0.0399 − 0.0399i)13-s + 0.831i·14-s + (1.02 − 0.541i)15-s − 0.0180·16-s + (0.296 + 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.967 + 0.253i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ -0.967 + 0.253i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0285833 - 0.222238i\)
\(L(\frac12)\) \(\approx\) \(0.0285833 - 0.222238i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.77 + 1.46i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 + (0.864 + 0.864i)T + 4iT^{2} \)
3 \( 1 + (2.45 - 2.45i)T - 9iT^{2} \)
7 \( 1 + (6.72 + 6.72i)T + 49iT^{2} \)
13 \( 1 + (-0.519 + 0.519i)T - 169iT^{2} \)
17 \( 1 + (-5.04 - 5.04i)T + 289iT^{2} \)
19 \( 1 + 25.5iT - 361T^{2} \)
23 \( 1 + (5.12 - 5.12i)T - 529iT^{2} \)
29 \( 1 - 0.0328iT - 841T^{2} \)
31 \( 1 + 51.6T + 961T^{2} \)
37 \( 1 + (17.2 + 17.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.0T + 1.68e3T^{2} \)
43 \( 1 + (-49.5 + 49.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (60.9 + 60.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-19.8 + 19.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 108. iT - 3.48e3T^{2} \)
61 \( 1 - 103.T + 3.72e3T^{2} \)
67 \( 1 + (-13.8 - 13.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 65.5T + 5.04e3T^{2} \)
73 \( 1 + (39.0 - 39.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 29.1iT - 6.24e3T^{2} \)
83 \( 1 + (-55.8 + 55.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 6.65iT - 7.92e3T^{2} \)
97 \( 1 + (-34.5 - 34.5i)T + 9.40e3iT^{2} \)
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

−14.86290137739138155552243778421, −13.29515505810438623522379704878, −11.79982164427181389041029411870, −10.89715325497930999782491066579, −10.13130569186413203864492271296, −8.966258078616924229922912789623, −7.02955156488428508665474948536, −5.36203099952563642872309806974, −3.88486998208537043868867333864, −0.26922146434579601786721871911, 3.44658421505604742158215086878, 6.03591269644729694775129567954, 6.98666149012794671939772920991, 8.120471924500554429036506050937, 9.466985049766085731114693360009, 11.39702728454446698401163882642, 12.33461723522029170180149439981, 12.71074445787843264415932834500, 14.73901181932060958000327951350, 16.00289099678856871470213797269

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