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Properties

Label 2-550-11.5-c1-0-13
Degree $2$
Conductor $550$
Sign $0.0219 + 0.999i$
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 + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (1.80 − 1.31i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 2.48i)9-s + (−1.23 − 3.07i)11-s − 0.618·12-s + (1.30 + 4.02i)13-s + (−1.80 − 1.31i)14-s + (0.309 − 0.951i)16-s + (0.809 − 2.48i)17-s + (−2.11 + 1.53i)18-s + (−0.309 − 0.224i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.288 + 0.209i)3-s + (−0.404 + 0.293i)4-s + (0.0779 − 0.239i)6-s + (0.683 − 0.496i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.829i)9-s + (−0.372 − 0.927i)11-s − 0.178·12-s + (0.363 + 1.11i)13-s + (−0.483 − 0.351i)14-s + (0.0772 − 0.237i)16-s + (0.196 − 0.603i)17-s + (−0.499 + 0.362i)18-s + (−0.0708 − 0.0515i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.0219 + 0.999i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985172 - 0.963747i\)
\(L(\frac12)\) \(\approx\) \(0.985172 - 0.963747i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good3 \( 1 + (-0.5 - 0.363i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.80 + 1.31i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.30 - 4.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.809 + 2.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.309 + 0.224i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (-7.66 + 5.56i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.78 + 5.48i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.73 + 1.26i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.85 + 5.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.527T + 43T^{2} \)
47 \( 1 + (-3.42 - 2.48i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.11 - 3.44i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.04 + 1.48i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.690 - 2.12i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.32T + 67T^{2} \)
71 \( 1 + (4.57 - 14.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.0 - 8.73i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.83 - 5.65i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.78 + 8.55i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.18T + 89T^{2} \)
97 \( 1 + (-3.70 - 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.63676265253664432663881169786, −9.694459773198906708735256592731, −8.867212636620179763009773907126, −8.225266920821470604339491579037, −7.08599327818047503478115493529, −5.92266679071401456802796133195, −4.59240063397054813432271120616, −3.71650288545644717929278058225, −2.56456482855910133228178817823, −0.909834619166162361229114892992, 1.66898707992793268170819857843, 3.07826092333088880951295018187, 4.84592775826124543848570205482, 5.31602000236144683602024260028, 6.58848741665435749858409045822, 7.63046634759462610059548962024, 8.259295774337207122779362967891, 8.871959323442526062742066184177, 10.25790449642380175191092203002, 10.67131162618513741580798975928

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