↑

Properties

Label 2-55-11.5-c1-0-2
Degree $2$
Conductor $55$
Sign $0.451 + 0.892i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  + (−0.579 − 1.78i)2-s + (1.43 + 1.04i)3-s + (−1.22 + 0.893i)4-s + (0.309 − 0.951i)5-s + (1.03 − 3.17i)6-s + (−3.44 + 2.50i)7-s + (−0.729 − 0.529i)8-s + (0.0492 + 0.151i)9-s − 1.87·10-s + (2.44 + 2.24i)11-s − 2.70·12-s + (0.420 + 1.29i)13-s + (6.46 + 4.69i)14-s + (1.43 − 1.04i)15-s + (−1.46 + 4.49i)16-s + (0.648 − 1.99i)17-s + ⋯
L(s)  = 1  + (−0.409 − 1.26i)2-s + (0.830 + 0.603i)3-s + (−0.614 + 0.446i)4-s + (0.138 − 0.425i)5-s + (0.420 − 1.29i)6-s + (−1.30 + 0.945i)7-s + (−0.257 − 0.187i)8-s + (0.0164 + 0.0505i)9-s − 0.593·10-s + (0.737 + 0.675i)11-s − 0.779·12-s + (0.116 + 0.358i)13-s + (1.72 + 1.25i)14-s + (0.371 − 0.269i)15-s + (−0.365 + 1.12i)16-s + (0.157 − 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705060 - 0.433197i\)
\(L(\frac12)\) \(\approx\) \(0.705060 - 0.433197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.44 - 2.24i)T \)
good2 \( 1 + (0.579 + 1.78i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.43 - 1.04i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (3.44 - 2.50i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.420 - 1.29i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.648 + 1.99i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.489 + 0.355i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (-5.36 + 3.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.678 + 2.08i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.99 - 3.62i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.99 + 4.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + (-2.48 - 1.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.05 + 6.32i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.73 - 7.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.75 - 5.41i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (1.61 - 4.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.584 + 0.424i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.75 - 5.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.294 - 0.906i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + (-3.56 - 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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

−15.28474377918376331182109105710, −13.86222345423448261895975376290, −12.44772389098754262565045882646, −11.90112766681534973153899704775, −10.06498780138290884972709111709, −9.433855269585725856846801809122, −8.761500535745882288904193326762, −6.32566307842299318672387587569, −3.91067239105012126372886748289, −2.54691041817812437494498123915, 3.28912258884919088878562794116, 6.14051789981519436023125711652, 7.00592571643802829310508230906, 8.059144934006416348291466706319, 9.191849421381954511482485537377, 10.58063373073910852841072329383, 12.52524046078440788046210220228, 13.86489649015460187158009118873, 14.25273394229447742968855979751, 15.69139696328869100198401947513

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