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Properties

Label 2-55-11.2-c2-0-2
Degree $2$
Conductor $55$
Sign $0.673 + 0.739i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.40 − 1.10i)2-s + (−3.07 + 2.23i)3-s + (7.11 + 5.16i)4-s + (−0.690 − 2.12i)5-s + (12.9 − 4.19i)6-s + (4.78 − 6.58i)7-s + (−10.0 − 13.8i)8-s + (1.67 − 5.15i)9-s + 7.99i·10-s + (7.96 − 7.58i)11-s − 33.3·12-s + (20.9 + 6.79i)13-s + (−23.5 + 17.1i)14-s + (6.87 + 4.99i)15-s + (8.07 + 24.8i)16-s + (0.261 − 0.0849i)17-s + ⋯
L(s)  = 1  + (−1.70 − 0.552i)2-s + (−1.02 + 0.744i)3-s + (1.77 + 1.29i)4-s + (−0.138 − 0.425i)5-s + (2.15 − 0.699i)6-s + (0.683 − 0.940i)7-s + (−1.25 − 1.73i)8-s + (0.186 − 0.573i)9-s + 0.799i·10-s + (0.724 − 0.689i)11-s − 2.78·12-s + (1.60 + 0.522i)13-s + (−1.68 + 1.22i)14-s + (0.458 + 0.332i)15-s + (0.504 + 1.55i)16-s + (0.0153 − 0.00499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.388122 - 0.171476i\)
\(L(\frac12)\) \(\approx\) \(0.388122 - 0.171476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 + (-7.96 + 7.58i)T \)
good2 \( 1 + (3.40 + 1.10i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (3.07 - 2.23i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-4.78 + 6.58i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-20.9 - 6.79i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-0.261 + 0.0849i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (9.69 + 13.3i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 1.44T + 529T^{2} \)
29 \( 1 + (-24.5 + 33.7i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-0.152 + 0.468i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-17.6 - 12.8i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-29.5 - 40.6i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 62.2iT - 1.84e3T^{2} \)
47 \( 1 + (10.9 - 7.94i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-1.14 + 3.52i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-7.28 - 5.29i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (78.3 - 25.4i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 36.5T + 4.48e3T^{2} \)
71 \( 1 + (-29.6 - 91.1i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-54.0 + 74.3i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (32.6 + 10.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (16.9 - 5.50i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 24.9T + 7.92e3T^{2} \)
97 \( 1 + (29.0 - 89.3i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−15.69810703907578137543848776165, −13.66894116987764217937985084180, −11.71560414705074123990276664077, −11.16205385078750279463003328156, −10.44823560711641212058195469383, −9.109331832553966072694825695874, −8.088615656008583006135202029825, −6.38893402101444723127795765848, −4.19768308509162719413946072869, −0.965149467627808304917763812519, 1.45601656647546670670135299283, 5.85631338385764292429693020540, 6.67457219709163743986137057019, 7.985209103822211211302269407054, 9.038989031574201683906556744726, 10.63931575582013890082092095722, 11.40185817597018250635761183582, 12.47074995199870384239491982284, 14.60538415604074023142527429508, 15.59219435584751185382630967461

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