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Properties

Label 2-55-11.4-c1-0-3
Degree $2$
Conductor $55$
Sign $0.739 + 0.673i$
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  + (−0.0756 − 0.0549i)2-s + (0.453 − 1.39i)3-s + (−0.615 − 1.89i)4-s + (−0.809 + 0.587i)5-s + (−0.110 + 0.0806i)6-s + (1.39 + 4.30i)7-s + (−0.115 + 0.354i)8-s + (0.686 + 0.498i)9-s + 0.0935·10-s + (−2.39 − 2.29i)11-s − 2.92·12-s + (0.924 + 0.671i)13-s + (0.130 − 0.402i)14-s + (0.453 + 1.39i)15-s + (−3.19 + 2.32i)16-s + (−2.72 + 1.98i)17-s + ⋯
L(s)  = 1  + (−0.0534 − 0.0388i)2-s + (0.261 − 0.805i)3-s + (−0.307 − 0.946i)4-s + (−0.361 + 0.262i)5-s + (−0.0452 + 0.0329i)6-s + (0.528 + 1.62i)7-s + (−0.0407 + 0.125i)8-s + (0.228 + 0.166i)9-s + 0.0295·10-s + (−0.723 − 0.690i)11-s − 0.843·12-s + (0.256 + 0.186i)13-s + (0.0349 − 0.107i)14-s + (0.117 + 0.360i)15-s + (−0.798 + 0.580i)16-s + (−0.661 + 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778152 - 0.301272i\)
\(L(\frac12)\) \(\approx\) \(0.778152 - 0.301272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 + 2.29i)T \)
good2 \( 1 + (0.0756 + 0.0549i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.453 + 1.39i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.39 - 4.30i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.924 - 0.671i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.72 - 1.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.88 + 5.78i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.45T + 23T^{2} \)
29 \( 1 + (-1.02 - 3.15i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.44 + 1.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.460 - 1.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.539 - 1.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.263T + 43T^{2} \)
47 \( 1 + (-2.13 + 6.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.16 - 0.846i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.18 + 6.72i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.02 - 1.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.516T + 67T^{2} \)
71 \( 1 + (-8.68 + 6.30i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.75 + 5.40i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.14 - 6.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.62 - 2.63i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-2.71 - 1.97i)T + (29.9 + 92.2i)T^{2} \)
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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

−15.24538030413445673011047167735, −14.04239886742272692210702624474, −13.09229833418332536575947354181, −11.76187859983127453172267019501, −10.70392185370278958484396884066, −9.054457426749347955547235919320, −8.102407547247770772874187873860, −6.39099728462163925674248056550, −5.08338178394585410132280103937, −2.22958235079510487272992591066, 3.78641443683664671578735000907, 4.55888984588800133593494241046, 7.30773922762058708654097723428, 8.128342189611578373152660761357, 9.668427875746510150716741589468, 10.66186999550995766035570286007, 12.12882296882788741867914895213, 13.29373242917554276270222390737, 14.29918338904356413980797634033, 15.73545980252126709203189529169

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