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Properties

Label 2-550-25.21-c1-0-17
Degree $2$
Conductor $550$
Sign $0.259 + 0.965i$
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  + (0.309 − 0.951i)2-s + (−0.150 − 0.109i)3-s + (−0.809 − 0.587i)4-s + (1.98 − 1.02i)5-s + (−0.150 + 0.109i)6-s + 4.69·7-s + (−0.809 + 0.587i)8-s + (−0.916 − 2.82i)9-s + (−0.362 − 2.20i)10-s + (−0.309 + 0.951i)11-s + (0.0573 + 0.176i)12-s + (1.83 + 5.65i)13-s + (1.45 − 4.46i)14-s + (−0.410 − 0.0625i)15-s + (0.309 + 0.951i)16-s + (0.958 − 0.696i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.0867 − 0.0630i)3-s + (−0.404 − 0.293i)4-s + (0.888 − 0.459i)5-s + (−0.0613 + 0.0445i)6-s + 1.77·7-s + (−0.286 + 0.207i)8-s + (−0.305 − 0.940i)9-s + (−0.114 − 0.697i)10-s + (−0.0931 + 0.286i)11-s + (0.0165 + 0.0509i)12-s + (0.509 + 1.56i)13-s + (0.388 − 1.19i)14-s + (−0.106 − 0.0161i)15-s + (0.0772 + 0.237i)16-s + (0.232 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59453 - 1.22208i\)
\(L(\frac12)\) \(\approx\) \(1.59453 - 1.22208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-1.98 + 1.02i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.150 + 0.109i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 4.69T + 7T^{2} \)
13 \( 1 + (-1.83 - 5.65i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.958 + 0.696i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.57 - 3.32i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.71 + 5.27i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.92 + 1.39i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.877 + 0.637i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.32 + 4.09i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.157 + 0.484i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (-1.52 - 1.10i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.55 + 6.21i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.71 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.33 + 4.12i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.41 - 5.38i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.44 + 1.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.74 - 14.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.7 - 8.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.396 - 0.287i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.48 - 4.56i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.69 + 4.13i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.76004601193255780935903955167, −9.787065784736559580727474506482, −8.822324874589764128300151323170, −8.372343800347292454413229941358, −6.78926298105473928506993852042, −5.82840932735656579766611378090, −4.79328346253023343898605200009, −4.05768747439442176998478985216, −2.19666804582242131113791866089, −1.38754231935700621957100388018, 1.73393412817229997231516367986, 3.14476326812701645219889280526, 4.87799983923654210437519578820, 5.30943409155836610707554846860, 6.23523508589354965128060795009, 7.56178824469147190046524750055, 8.134453365826880470954931867157, 8.920878469638180949138199311138, 10.36757781857912354680341463125, 10.83322618276583274327710702478

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