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Properties

Label 2-55-55.32-c5-0-22
Degree $2$
Conductor $55$
Sign $0.389 + 0.921i$
Analytic cond. $8.82111$
Root an. cond. $2.97003$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.43 + 4.43i)2-s + (16.3 − 16.3i)3-s − 7.25i·4-s + (27.0 − 48.9i)5-s + 144. i·6-s + (30.3 − 30.3i)7-s + (−109. − 109. i)8-s − 291. i·9-s + (96.9 + 336. i)10-s + (−376. + 138. i)11-s + (−118. − 118. i)12-s + (−654. − 654. i)13-s + 268. i·14-s + (−357. − 1.24e3i)15-s + 1.20e3·16-s + (607. − 607. i)17-s + ⋯
L(s)  = 1  + (−0.783 + 0.783i)2-s + (1.04 − 1.04i)3-s − 0.226i·4-s + (0.483 − 0.875i)5-s + 1.64i·6-s + (0.234 − 0.234i)7-s + (−0.605 − 0.605i)8-s − 1.20i·9-s + (0.306 + 1.06i)10-s + (−0.938 + 0.344i)11-s + (−0.237 − 0.237i)12-s + (−1.07 − 1.07i)13-s + 0.366i·14-s + (−0.410 − 1.42i)15-s + 1.17·16-s + (0.509 − 0.509i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.389 + 0.921i$
Analytic conductor: \(8.82111\)
Root analytic conductor: \(2.97003\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :5/2),\ 0.389 + 0.921i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.19997 - 0.795732i\)
\(L(\frac12)\) \(\approx\) \(1.19997 - 0.795732i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-27.0 + 48.9i)T \)
11 \( 1 + (376. - 138. i)T \)
good2 \( 1 + (4.43 - 4.43i)T - 32iT^{2} \)
3 \( 1 + (-16.3 + 16.3i)T - 243iT^{2} \)
7 \( 1 + (-30.3 + 30.3i)T - 1.68e4iT^{2} \)
13 \( 1 + (654. + 654. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-607. + 607. i)T - 1.41e6iT^{2} \)
19 \( 1 - 2.32e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.24e3 + 1.24e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 5.39e3T + 2.05e7T^{2} \)
31 \( 1 - 588.T + 2.86e7T^{2} \)
37 \( 1 + (-1.22e3 - 1.22e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.22e4iT - 1.15e8T^{2} \)
43 \( 1 + (-6.02e3 - 6.02e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.48e4 - 1.48e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (-1.12e4 + 1.12e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 1.53e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.86e4iT - 8.44e8T^{2} \)
67 \( 1 + (1.88e3 + 1.88e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 7.82e4T + 1.80e9T^{2} \)
73 \( 1 + (-5.45e4 - 5.45e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 6.18e4T + 3.07e9T^{2} \)
83 \( 1 + (-144. - 144. i)T + 3.93e9iT^{2} \)
89 \( 1 + 4.73e3iT - 5.58e9T^{2} \)
97 \( 1 + (1.48e4 + 1.48e4i)T + 8.58e9iT^{2} \)
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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

−14.08703804949664991482289314567, −12.97290759160814744632026410387, −12.35762603774260098531107050426, −9.931580658416968551304716806351, −8.956415263418457531173881625948, −7.71535339460134316339737806901, −7.45009590382240434545583641124, −5.41886642889066608759957306015, −2.74267377527206536719333413724, −0.815622565749531125959866347103, 2.17815587348322410901939392389, 3.26069373599558554389250825879, 5.38145530395551961583220985017, 7.65307220784955817468728969246, 9.122651290150726769758414448496, 9.766689566116049499918868729007, 10.60874033207745869445867583192, 11.72961120472558614836982933016, 13.73522970552454295201671749565, 14.63551755300447480899391977531

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