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Properties

Label 2-550-55.4-c1-0-15
Degree $2$
Conductor $550$
Sign $0.945 + 0.325i$
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 + (3.20 + 1.04i)3-s + (−0.309 − 0.951i)4-s + (2.72 − 1.98i)6-s + (0.177 − 0.0577i)7-s + (−0.951 − 0.309i)8-s + (6.77 + 4.92i)9-s + (−0.359 − 3.29i)11-s − 3.37i·12-s + (−1.79 + 2.46i)13-s + (0.0577 − 0.177i)14-s + (−0.809 + 0.587i)16-s + (−1.73 − 2.38i)17-s + (7.96 − 2.58i)18-s + (−1.66 + 5.11i)19-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (1.85 + 0.601i)3-s + (−0.154 − 0.475i)4-s + (1.11 − 0.809i)6-s + (0.0672 − 0.0218i)7-s + (−0.336 − 0.109i)8-s + (2.25 + 1.64i)9-s + (−0.108 − 0.994i)11-s − 0.973i·12-s + (−0.497 + 0.684i)13-s + (0.0154 − 0.0475i)14-s + (−0.202 + 0.146i)16-s + (−0.420 − 0.578i)17-s + (1.87 − 0.610i)18-s + (−0.381 + 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05029 - 0.509790i\)
\(L(\frac12)\) \(\approx\) \(3.05029 - 0.509790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.359 + 3.29i)T \)
good3 \( 1 + (-3.20 - 1.04i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.177 + 0.0577i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.79 - 2.46i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.73 + 2.38i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.66 - 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.115iT - 23T^{2} \)
29 \( 1 + (3.14 + 9.67i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.22 + 4.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.41 + 2.08i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.44 - 4.45i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.27iT - 43T^{2} \)
47 \( 1 + (-6.04 - 1.96i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.73 - 6.51i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.479 - 1.47i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.711 - 0.516i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.03iT - 67T^{2} \)
71 \( 1 + (-0.533 + 0.387i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.21 + 0.720i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.45 + 5.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.86 - 2.57i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.69T + 89T^{2} \)
97 \( 1 + (-5.20 + 7.16i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.64883161455327042426100107636, −9.584912909144362188124380194422, −9.294192457054655209820369254349, −8.191480205024924432817173291278, −7.54182716074316159518056374914, −6.01142650411619898360078255878, −4.54590132400195117287454569878, −3.89203214048352041668729428001, −2.85582070503456579755333909846, −1.94566010348849031796983271754, 1.91186523381395840065117814132, 2.95594859978079807858206327290, 3.98816986756747554714665213168, 5.10347959534660326082240831930, 6.82163148966616368332102807742, 7.18915932545896040226826545007, 8.112562771279229956167250763947, 8.861017209345158882440331937319, 9.546790435453393141899019911100, 10.65602335380081887802133801225

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