Properties

Label 2-55-55.4-c1-0-0
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: $\chi_{55} (4, \cdot )$
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

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

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