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Properties

Label 2-55-11.3-c1-0-2
Degree $2$
Conductor $55$
Sign $0.475 + 0.879i$
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  + (−2.04 + 1.48i)2-s + (−0.762 − 2.34i)3-s + (1.35 − 4.15i)4-s + (−0.809 − 0.587i)5-s + (5.03 + 3.66i)6-s + (0.646 − 1.99i)7-s + (1.85 + 5.69i)8-s + (−2.49 + 1.81i)9-s + 2.52·10-s + (−1.64 − 2.87i)11-s − 10.7·12-s + (−1.04 + 0.757i)13-s + (1.63 + 5.02i)14-s + (−0.762 + 2.34i)15-s + (−5.16 − 3.74i)16-s + (2.41 + 1.75i)17-s + ⋯
L(s)  = 1  + (−1.44 + 1.04i)2-s + (−0.440 − 1.35i)3-s + (0.675 − 2.07i)4-s + (−0.361 − 0.262i)5-s + (2.05 + 1.49i)6-s + (0.244 − 0.752i)7-s + (0.654 + 2.01i)8-s + (−0.831 + 0.604i)9-s + 0.798·10-s + (−0.496 − 0.867i)11-s − 3.11·12-s + (−0.289 + 0.210i)13-s + (0.436 + 1.34i)14-s + (−0.196 + 0.605i)15-s + (−1.29 − 0.937i)16-s + (0.586 + 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.475 + 0.879i$
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: \((2,\ 55,\ (\ :1/2),\ 0.475 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310957 - 0.185327i\)
\(L(\frac12)\) \(\approx\) \(0.310957 - 0.185327i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 2.87i)T \)
good2 \( 1 + (2.04 - 1.48i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.762 + 2.34i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.646 + 1.99i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.04 - 0.757i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.41 - 1.75i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.664 - 2.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.77T + 23T^{2} \)
29 \( 1 + (0.189 - 0.582i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.94 + 2.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.578 - 1.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.57 + 4.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + (2.25 + 6.94i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.38 + 1.72i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.00 - 6.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.406 - 0.295i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 + (-9.14 - 6.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.43 - 10.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.33 - 3.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.77 + 6.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (-0.284 + 0.206i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.51192932802303339565236077463, −14.18899932050607487478237314669, −12.95577953577690869872745797815, −11.47693756877290649411527535529, −10.34205557946897595917958140710, −8.675717682552939395549971323265, −7.68749014810315208960676636727, −6.96621095453793822412819037282, −5.64835807746032859473021989752, −0.984038280075195917872007938389, 2.95266457225367668719649832187, 4.91015182859007254610520727855, 7.50202630841399602285424976323, 8.960086115307323853636824467050, 9.806889062704875524889959937534, 10.73214846330112607453475702167, 11.53927042439094831268561555887, 12.58227100057270409469063723575, 14.97289886759718218346586198937, 15.76416426504392425086603432913

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