↑

Properties

Label 2-55-55.32-c5-0-0
Degree $2$
Conductor $55$
Sign $-0.961 + 0.276i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.819 + 0.819i)2-s + (−2.25 + 2.25i)3-s + 30.6i·4-s + (5.74 − 55.6i)5-s − 3.69i·6-s + (−88.0 + 88.0i)7-s + (−51.3 − 51.3i)8-s + 232. i·9-s + (40.8 + 50.2i)10-s + (68.0 − 395. i)11-s + (−69.1 − 69.1i)12-s + (−741. − 741. i)13-s − 144. i·14-s + (112. + 138. i)15-s − 896.·16-s + (−805. + 805. i)17-s + ⋯
L(s)  = 1  + (−0.144 + 0.144i)2-s + (−0.144 + 0.144i)3-s + 0.957i·4-s + (0.102 − 0.994i)5-s − 0.0419i·6-s + (−0.679 + 0.679i)7-s + (−0.283 − 0.283i)8-s + 0.958i·9-s + (0.129 + 0.159i)10-s + (0.169 − 0.985i)11-s + (−0.138 − 0.138i)12-s + (−1.21 − 1.21i)13-s − 0.196i·14-s + (0.129 + 0.158i)15-s − 0.875·16-s + (−0.676 + 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.961 + 0.276i$
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: \((2,\ 55,\ (\ :5/2),\ -0.961 + 0.276i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0264493 - 0.187577i\)
\(L(\frac12)\) \(\approx\) \(0.0264493 - 0.187577i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.74 + 55.6i)T \)
11 \( 1 + (-68.0 + 395. i)T \)
good2 \( 1 + (0.819 - 0.819i)T - 32iT^{2} \)
3 \( 1 + (2.25 - 2.25i)T - 243iT^{2} \)
7 \( 1 + (88.0 - 88.0i)T - 1.68e4iT^{2} \)
13 \( 1 + (741. + 741. i)T + 3.71e5iT^{2} \)
17 \( 1 + (805. - 805. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.85e3T + 2.47e6T^{2} \)
23 \( 1 + (561. - 561. i)T - 6.43e6iT^{2} \)
29 \( 1 - 3.98e3T + 2.05e7T^{2} \)
31 \( 1 - 4.01e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.64e3 - 5.64e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.42e4iT - 1.15e8T^{2} \)
43 \( 1 + (-3.12e3 - 3.12e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.84e4 + 1.84e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (2.40e3 - 2.40e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.28e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.78e4iT - 8.44e8T^{2} \)
67 \( 1 + (-5.68e3 - 5.68e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 3.27e3T + 1.80e9T^{2} \)
73 \( 1 + (2.58e4 + 2.58e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 4.13e4T + 3.07e9T^{2} \)
83 \( 1 + (6.78e4 + 6.78e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 8.65e4iT - 5.58e9T^{2} \)
97 \( 1 + (2.45e4 + 2.45e4i)T + 8.58e9iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.24241573446063007437666976109, −13.32503646263228648261462461668, −12.79363719978664665151077173529, −11.71686262743589927978632017160, −10.12358663225437473086886444661, −8.692828501328047002441157706770, −8.008266541557588076814027887052, −6.11042317784967262932971514106, −4.61367435308821549674070413295, −2.71606001028200481415376098471, 0.091784394914907296247827758224, 2.25892657862403954412163010807, 4.38393614582671776900263986046, 6.52501547364031307520393012923, 6.91059065965029127994163258163, 9.417598396425418390920324071145, 10.01117683908447520129605344734, 11.22428112493152648863397927051, 12.41509752573278471012143175044, 14.01174999047620513280946667744

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