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Properties

Label 2-550-5.4-c1-0-0
Degree $2$
Conductor $550$
Sign $-0.894 - 0.447i$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
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  − i·2-s + 3.37i·3-s − 4-s + 3.37·6-s + 3.37i·7-s + i·8-s − 8.37·9-s − 11-s − 3.37i·12-s − 2i·13-s + 3.37·14-s + 16-s + 1.37i·17-s + 8.37i·18-s − 0.627·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.94i·3-s − 0.5·4-s + 1.37·6-s + 1.27i·7-s + 0.353i·8-s − 2.79·9-s − 0.301·11-s − 0.973i·12-s − 0.554i·13-s + 0.901·14-s + 0.250·16-s + 0.332i·17-s + 1.97i·18-s − 0.144·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203088 + 0.860297i\)
\(L(\frac12)\) \(\approx\) \(0.203088 + 0.860297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 3.37iT - 3T^{2} \)
7 \( 1 - 3.37iT - 7T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 1.37iT - 17T^{2} \)
19 \( 1 + 0.627T + 19T^{2} \)
23 \( 1 + 2.74iT - 23T^{2} \)
29 \( 1 + 1.37T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 - 9.37iT - 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2.74iT - 47T^{2} \)
53 \( 1 - 4.11iT - 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 15.4iT - 73T^{2} \)
79 \( 1 - 1.25T + 79T^{2} \)
83 \( 1 - 2.74iT - 83T^{2} \)
89 \( 1 - 1.37T + 89T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−11.04902827698842324794549119004, −10.14750291407820377044346853101, −9.735846337696146963833564300828, −8.650983815026329647850826620367, −8.352127217857338524142016911424, −6.10644110922531971354816751195, −5.26322897283354987904118940913, −4.54581059174743706305447595745, −3.35163103565771022975707468936, −2.57061996064663028381270237007, 0.49767282219092026449500653311, 1.90194442575267211263859162824, 3.52911694542513406458220473280, 5.08918362404901425834331760323, 6.21181193432819989212695298680, 6.97763241348462825630126225871, 7.47323918995497605882606571381, 8.214659649000683756938393505544, 9.189823413623371348719690308983, 10.51945937914225155050407751508

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