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Properties

Label 2-550-55.14-c1-0-6
Degree $2$
Conductor $550$
Sign $0.994 + 0.105i$
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  + (−0.587 − 0.809i)2-s + (0.363 − 0.118i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.224i)6-s + (2.85 + 0.927i)7-s + (0.951 − 0.309i)8-s + (−2.30 + 1.67i)9-s + (−1.23 − 3.07i)11-s + 0.381i·12-s + (3.66 + 5.04i)13-s + (−0.927 − 2.85i)14-s + (−0.809 − 0.587i)16-s + (2.57 − 3.54i)17-s + (2.71 + 0.881i)18-s + (1.80 + 5.56i)19-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.209 − 0.0681i)3-s + (−0.154 + 0.475i)4-s + (−0.126 − 0.0916i)6-s + (1.07 + 0.350i)7-s + (0.336 − 0.109i)8-s + (−0.769 + 0.559i)9-s + (−0.372 − 0.927i)11-s + 0.110i·12-s + (1.01 + 1.39i)13-s + (−0.247 − 0.762i)14-s + (−0.202 − 0.146i)16-s + (0.624 − 0.859i)17-s + (0.639 + 0.207i)18-s + (0.415 + 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34804 - 0.0711868i\)
\(L(\frac12)\) \(\approx\) \(1.34804 - 0.0711868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good3 \( 1 + (-0.363 + 0.118i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-2.85 - 0.927i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.66 - 5.04i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.57 + 3.54i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.80 - 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.85iT - 23T^{2} \)
29 \( 1 + (-0.163 + 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.0 - 3.26i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.14 - 3.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + (-1.40 + 0.454i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.16 - 4.35i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.07 + 6.37i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.04 + 5.11i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.0901iT - 67T^{2} \)
71 \( 1 + (-10.7 - 7.83i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (10.9 + 3.57i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.28 - 4.56i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.21 + 3.04i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (-0.555 - 0.763i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.07891789509792809809197423421, −9.885206140496353370004849663785, −8.893531146345398435850350190379, −8.254823100814859177156837879702, −7.62623672323635243217798513122, −6.07578560925228361386429082190, −5.15669081082101869389548425409, −3.85964921340142797049913979302, −2.64590072389035433314594575856, −1.42174216129403381702842019037, 1.05726982008618626402353127153, 2.83436660016556518220606332008, 4.29636258678203687898325710414, 5.36265172213007173423569758795, 6.21892500285645456081578266463, 7.49784269302349926812671830843, 8.098395084015489458203561130522, 8.817840247474573282782602488876, 9.873001090101870054924531527127, 10.74688773854449945343715298933

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