↑

Properties

Label 2-55-55.13-c1-0-2
Degree $2$
Conductor $55$
Sign $0.877 - 0.479i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.513 + 0.261i)2-s + (0.760 + 0.120i)3-s + (−0.980 + 1.34i)4-s + (2.23 + 0.0653i)5-s + (−0.421 + 0.136i)6-s + (−0.186 − 1.17i)7-s + (0.330 − 2.08i)8-s + (−2.28 − 0.743i)9-s + (−1.16 + 0.550i)10-s + (−0.502 + 3.27i)11-s + (−0.908 + 0.908i)12-s + (−2.75 − 5.41i)13-s + (0.403 + 0.555i)14-s + (1.69 + 0.318i)15-s + (−0.655 − 2.01i)16-s + (0.168 − 0.330i)17-s + ⋯
L(s)  = 1  + (−0.362 + 0.184i)2-s + (0.438 + 0.0695i)3-s + (−0.490 + 0.674i)4-s + (0.999 + 0.0292i)5-s + (−0.172 + 0.0559i)6-s + (−0.0705 − 0.445i)7-s + (0.116 − 0.737i)8-s + (−0.763 − 0.247i)9-s + (−0.368 + 0.174i)10-s + (−0.151 + 0.988i)11-s + (−0.262 + 0.262i)12-s + (−0.765 − 1.50i)13-s + (0.107 + 0.148i)14-s + (0.436 + 0.0823i)15-s + (−0.163 − 0.504i)16-s + (0.0408 − 0.0801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767489 + 0.195867i\)
\(L(\frac12)\) \(\approx\) \(0.767489 + 0.195867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.0653i)T \)
11 \( 1 + (0.502 - 3.27i)T \)
good2 \( 1 + (0.513 - 0.261i)T + (1.17 - 1.61i)T^{2} \)
3 \( 1 + (-0.760 - 0.120i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (0.186 + 1.17i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (2.75 + 5.41i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.168 + 0.330i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.11 - 0.809i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \)
29 \( 1 + (-4.95 - 3.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.764 - 2.35i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.93 - 0.465i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.60 + 4.95i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.75 - 6.75i)T - 43iT^{2} \)
47 \( 1 + (0.199 - 1.26i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.363 - 0.185i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.110 - 0.151i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.649 - 0.211i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \)
71 \( 1 + (-0.319 - 0.983i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.51 - 0.715i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.59 - 11.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-10.1 - 5.16i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + 8.04iT - 89T^{2} \)
97 \( 1 + (3.83 + 7.52i)T + (-57.0 + 78.4i)T^{2} \)
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

−15.33665710256822567967228509308, −14.25666724935322634046424420542, −13.21267599197319831998639991783, −12.35018312988895599195886852198, −10.35507176867738924062927099307, −9.484992858112086209732425645312, −8.310391446766748147252574655422, −7.06164199751535296062940296713, −5.14512056402110959283836608049, −3.06776956194270685861345794717, 2.33585851796872187937261904254, 5.07695155006679707522361662820, 6.33697028836350616658316391568, 8.574569381147008223696247611439, 9.192513447439731936749690187327, 10.38967898743068641538431323318, 11.63697805215827628259738498649, 13.40463427409499280812466599298, 14.08178260826312105661652051769, 14.86637735715550441357073181691

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