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Properties

Label 2-55-55.14-c1-0-2
Degree $2$
Conductor $55$
Sign $0.245 + 0.969i$
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  + (−0.725 − 0.998i)2-s + (−0.346 + 0.112i)3-s + (0.147 − 0.453i)4-s + (0.0238 − 2.23i)5-s + (0.363 + 0.264i)6-s + (2.45 + 0.798i)7-s + (−2.90 + 0.944i)8-s + (−2.31 + 1.68i)9-s + (−2.24 + 1.59i)10-s + (3.12 + 1.12i)11-s + 0.173i·12-s + (1.62 + 2.23i)13-s + (−0.985 − 3.03i)14-s + (0.243 + 0.776i)15-s + (2.27 + 1.65i)16-s + (−2.26 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.512 − 0.705i)2-s + (−0.199 + 0.0649i)3-s + (0.0737 − 0.226i)4-s + (0.0106 − 0.999i)5-s + (0.148 + 0.107i)6-s + (0.929 + 0.301i)7-s + (−1.02 + 0.333i)8-s + (−0.773 + 0.561i)9-s + (−0.711 + 0.505i)10-s + (0.940 + 0.339i)11-s + 0.0501i·12-s + (0.449 + 0.619i)13-s + (−0.263 − 0.810i)14-s + (0.0628 + 0.200i)15-s + (0.569 + 0.414i)16-s + (−0.549 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537278 - 0.418191i\)
\(L(\frac12)\) \(\approx\) \(0.537278 - 0.418191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0238 + 2.23i)T \)
11 \( 1 + (-3.12 - 1.12i)T \)
good2 \( 1 + (0.725 + 0.998i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.346 - 0.112i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-2.45 - 0.798i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.62 - 2.23i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.26 - 3.11i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0857 - 0.264i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.40iT - 23T^{2} \)
29 \( 1 + (1.02 - 3.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.456 - 0.331i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.497 - 0.161i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.57 + 4.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.54iT - 43T^{2} \)
47 \( 1 + (4.68 - 1.52i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.12 - 7.05i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.31 - 7.13i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.4 + 8.33i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.20iT - 67T^{2} \)
71 \( 1 + (-6.79 - 4.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (12.3 + 4.02i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.85 + 5.70i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.93 + 2.66i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 + (6.40 + 8.81i)T + (-29.9 + 92.2i)T^{2} \)
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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

−14.99985509861852945895134498871, −14.06990911915570951937399001024, −12.40289482625520856623690217809, −11.53657428751053321174025869808, −10.64070313557157476748871117909, −9.054205466256559674275725971206, −8.449211174199302255754010612488, −6.10874749854609150362245226240, −4.66639998614416985705782401760, −1.80567855218291239778268290432, 3.40210840687127325413544219627, 5.92194244313899952426756250319, 7.05469794989621774496536166490, 8.178708233557622709371050113487, 9.437744752438169856331928753427, 11.24142654725924826127867066251, 11.71964908879812819489883028120, 13.64579366618590351485027552149, 14.72243232705206917055039392514, 15.51318585884385469720715893123

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