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Properties

Label 2-55-55.2-c1-0-2
Degree $2$
Conductor $55$
Sign $0.661 + 0.749i$
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  + (−0.261 − 0.513i)2-s + (0.120 − 0.760i)3-s + (0.980 − 1.34i)4-s + (−1.76 + 1.36i)5-s + (−0.421 + 0.136i)6-s + (1.17 − 0.186i)7-s + (−2.08 − 0.330i)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 + (−5.41 + 2.75i)13-s + (−0.403 − 0.555i)14-s + (0.825 + 1.51i)15-s + (−0.655 − 2.01i)16-s + (0.330 + 0.168i)17-s + ⋯
L(s)  = 1  + (−0.184 − 0.362i)2-s + (0.0695 − 0.438i)3-s + (0.490 − 0.674i)4-s + (−0.791 + 0.611i)5-s + (−0.172 + 0.0559i)6-s + (0.445 − 0.0705i)7-s + (−0.737 − 0.116i)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 + (−1.50 + 0.765i)13-s + (−0.107 − 0.148i)14-s + (0.213 + 0.389i)15-s + (−0.163 − 0.504i)16-s + (0.0801 + 0.0408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.661 + 0.749i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.661 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738277 - 0.333224i\)
\(L(\frac12)\) \(\approx\) \(0.738277 - 0.333224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.76 - 1.36i)T \)
11 \( 1 + (0.502 - 3.27i)T \)
good2 \( 1 + (0.261 + 0.513i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (-0.120 + 0.760i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (-1.17 + 0.186i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (5.41 - 2.75i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.330 - 0.168i)T + (9.99 + 13.7i)T^{2} \)
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 \( 1 + (0.465 + 2.93i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (3.60 + 4.95i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.75 - 6.75i)T + 43iT^{2} \)
47 \( 1 + (1.26 + 0.199i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.185 - 0.363i)T + (-31.1 + 42.8i)T^{2} \)
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 \( 1 + (-0.715 - 4.51i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.59 + 11.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.16 + 10.1i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 8.04iT - 89T^{2} \)
97 \( 1 + (-7.52 + 3.83i)T + (57.0 - 78.4i)T^{2} \)
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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

−15.00597945975132628554837847930, −14.40296267572224139518533038774, −12.60226283157882052521678759037, −11.72375437909449330904455988577, −10.61283937000209686574001720657, −9.575765427438229473013973658590, −7.53483875237760741378546355654, −6.87470734013728367665563832871, −4.68686299969765933047277472462, −2.23701452163949610681131906161, 3.45552171887150784266093300837, 5.15726334648729579152720252335, 7.25032399552082226870441340822, 8.107058564439821617868113746097, 9.381735464886320909418381026164, 11.02380155404627652560807891873, 12.09851512397631091060124262723, 12.99352743719574080358343410173, 14.88101611477977938100491300300, 15.55196528483753144229930700224

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