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Properties

Label 2-55-11.3-c1-0-0
Degree $2$
Conductor $55$
Sign $0.171 - 0.985i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 + 0.796i)2-s + (0.177 + 0.547i)3-s + (−0.0501 + 0.154i)4-s + (0.809 + 0.587i)5-s + (−0.631 − 0.458i)6-s + (−1.12 + 3.47i)7-s + (−0.905 − 2.78i)8-s + (2.15 − 1.56i)9-s − 1.35·10-s + (0.490 − 3.28i)11-s − 0.0933·12-s + (2.29 − 1.66i)13-s + (−1.52 − 4.70i)14-s + (−0.177 + 0.547i)15-s + (2.95 + 2.14i)16-s + (−2.98 − 2.17i)17-s + ⋯
L(s)  = 1  + (−0.775 + 0.563i)2-s + (0.102 + 0.315i)3-s + (−0.0250 + 0.0771i)4-s + (0.361 + 0.262i)5-s + (−0.257 − 0.187i)6-s + (−0.426 + 1.31i)7-s + (−0.320 − 0.985i)8-s + (0.719 − 0.522i)9-s − 0.428·10-s + (0.147 − 0.989i)11-s − 0.0269·12-s + (0.635 − 0.461i)13-s + (−0.408 − 1.25i)14-s + (−0.0459 + 0.141i)15-s + (0.738 + 0.536i)16-s + (−0.724 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.171 - 0.985i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.171 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479979 + 0.403726i\)
\(L(\frac12)\) \(\approx\) \(0.479979 + 0.403726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.490 + 3.28i)T \)
good2 \( 1 + (1.09 - 0.796i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.177 - 0.547i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.12 - 3.47i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.29 + 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.98 + 2.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0293 + 0.0904i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + (2.08 - 6.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.04 + 9.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 + (0.687 + 2.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.42 - 1.75i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.62 - 8.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (6.71 + 4.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.407 - 1.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.2 + 8.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.61 - 6.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-3.50 + 2.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.83258467457267422897578303185, −14.90549873133334893468533118639, −13.27281463933113945225445130951, −12.28782956721667160495455103099, −10.68064969677109436120955395071, −9.173186710950987872323398616993, −8.834432340487950173439871754472, −7.04573961343914455254838720674, −5.82002225618961629837001933217, −3.38011110982820984564399557256, 1.67710591377061344305854649629, 4.45202187160724167795555229680, 6.58257037796091564409544749849, 7.965303768918533880529364739813, 9.512370072607687738969375209085, 10.22690996902973837176317538814, 11.31509395267830031126527058608, 12.99543263963177674474085748186, 13.66372567507367499873503977016, 15.04744088885958135691977307192

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