↑

Properties

Label 2-55-11.10-c2-0-4
Degree $2$
Conductor $55$
Sign $0.535 + 0.844i$
Analytic cond. $1.49864$
Root an. cond. $1.22419$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.46i·2-s + 0.114·3-s + 1.86·4-s + 2.23·5-s − 0.167i·6-s − 4.56i·7-s − 8.56i·8-s − 8.98·9-s − 3.26i·10-s + (5.89 + 9.28i)11-s + 0.213·12-s + 16.5i·13-s − 6.66·14-s + 0.256·15-s − 5.05·16-s − 17.2i·17-s + ⋯
L(s)  = 1  − 0.730i·2-s + 0.0381·3-s + 0.466·4-s + 0.447·5-s − 0.0278i·6-s − 0.651i·7-s − 1.07i·8-s − 0.998·9-s − 0.326i·10-s + (0.535 + 0.844i)11-s + 0.0178·12-s + 1.27i·13-s − 0.475·14-s + 0.0170·15-s − 0.316·16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17395 - 0.645527i\)
\(L(\frac12)\) \(\approx\) \(1.17395 - 0.645527i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23T \)
11 \( 1 + (-5.89 - 9.28i)T \)
good2 \( 1 + 1.46iT - 4T^{2} \)
3 \( 1 - 0.114T + 9T^{2} \)
7 \( 1 + 4.56iT - 49T^{2} \)
13 \( 1 - 16.5iT - 169T^{2} \)
17 \( 1 + 17.2iT - 289T^{2} \)
19 \( 1 - 35.8iT - 361T^{2} \)
23 \( 1 + 29.3T + 529T^{2} \)
29 \( 1 + 8.51iT - 841T^{2} \)
31 \( 1 + 26.3T + 961T^{2} \)
37 \( 1 - 44.4T + 1.36e3T^{2} \)
41 \( 1 + 52.2iT - 1.68e3T^{2} \)
43 \( 1 + 6.77iT - 1.84e3T^{2} \)
47 \( 1 - 15.0T + 2.20e3T^{2} \)
53 \( 1 + 33.1T + 2.80e3T^{2} \)
59 \( 1 - 51.5T + 3.48e3T^{2} \)
61 \( 1 + 23.1iT - 3.72e3T^{2} \)
67 \( 1 + 113.T + 4.48e3T^{2} \)
71 \( 1 - 8.00T + 5.04e3T^{2} \)
73 \( 1 + 32.5iT - 5.32e3T^{2} \)
79 \( 1 + 52.0iT - 6.24e3T^{2} \)
83 \( 1 - 43.3iT - 6.88e3T^{2} \)
89 \( 1 - 73.8T + 7.92e3T^{2} \)
97 \( 1 - 22.0T + 9.40e3T^{2} \)
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

−14.57000515499014124995599328951, −13.84579412680982573285077653821, −12.26338763133091248962446924422, −11.56688315435591868077543148778, −10.28596601990578832311633424353, −9.327769931827861834775259070508, −7.43969702475584404203980853587, −6.12679230052008586798972981634, −3.93831125514143753622104466982, −1.99565778490526600045696685395, 2.75584136954852601515315071505, 5.54713345154897930472551426192, 6.27908360837472805645640476979, 8.015837218665088524393195647623, 8.966561144628718457845937824634, 10.77666589082309009433420756659, 11.70402536067062103735555902640, 13.18446145816026083388920711875, 14.46523059179520259584166715042, 15.20109975396267457968008651279

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