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Properties

Label 2-55-11.5-c1-0-0
Degree $2$
Conductor $55$
Sign $0.0851 - 0.996i$
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  + (0.697 + 2.14i)2-s + (−0.628 − 0.456i)3-s + (−2.50 + 1.82i)4-s + (0.309 − 0.951i)5-s + (0.542 − 1.66i)6-s + (−0.100 + 0.0728i)7-s + (−2.00 − 1.45i)8-s + (−0.740 − 2.27i)9-s + 2.25·10-s + (−0.899 − 3.19i)11-s + 2.40·12-s + (1.69 + 5.22i)13-s + (−0.226 − 0.164i)14-s + (−0.628 + 0.456i)15-s + (−0.184 + 0.566i)16-s + (0.160 − 0.494i)17-s + ⋯
L(s)  = 1  + (0.493 + 1.51i)2-s + (−0.363 − 0.263i)3-s + (−1.25 + 0.910i)4-s + (0.138 − 0.425i)5-s + (0.221 − 0.681i)6-s + (−0.0379 + 0.0275i)7-s + (−0.709 − 0.515i)8-s + (−0.246 − 0.759i)9-s + 0.714·10-s + (−0.271 − 0.962i)11-s + 0.695·12-s + (0.470 + 1.44i)13-s + (−0.0605 − 0.0439i)14-s + (−0.162 + 0.117i)15-s + (−0.0460 + 0.141i)16-s + (0.0389 − 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.0851 - 0.996i$
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: \((2,\ 55,\ (\ :1/2),\ 0.0851 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690106 + 0.633675i\)
\(L(\frac12)\) \(\approx\) \(0.690106 + 0.633675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.899 + 3.19i)T \)
good2 \( 1 + (-0.697 - 2.14i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.628 + 0.456i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.100 - 0.0728i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.69 - 5.22i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.160 + 0.494i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.55 + 1.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + (-3.29 + 2.39i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.17 - 6.70i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.10 + 5.16i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.10 - 4.43i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (0.369 + 0.268i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0109 - 0.0337i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.42 + 3.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.37 + 7.31i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + (-3.79 + 11.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.89 - 5.00i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.93 - 5.96i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.193 + 0.595i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (0.567 + 1.74i)T + (-78.4 + 57.0i)T^{2} \)
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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

−15.86939656166084206603023070761, −14.41569646073921159194903023922, −13.75862915244577407546812506468, −12.53987093821885259938117135949, −11.26966056705669241131804940693, −9.195103899906610841084658852914, −8.156149061184127198034574043138, −6.57593427990265695386227359779, −5.88842260242961702141504186232, −4.26370428509084885869492079320, 2.46353696021421585871651661305, 4.22664295371424507929169692566, 5.71379921997527799409118307459, 7.937053824564291706663725917732, 10.03781571287046251319591629114, 10.45175197486914553598723219502, 11.57008253435964467006764239703, 12.68468734144618171619578504779, 13.55743276629324223939653622966, 14.78526946303714637514524313179

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