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Properties

Label 2-550-275.86-c1-0-23
Degree $2$
Conductor $550$
Sign $0.353 + 0.935i$
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 + (−0.805 − 2.47i)3-s + (−0.809 + 0.587i)4-s + (1.01 − 1.99i)5-s + (2.10 − 1.53i)6-s + (2.12 + 1.54i)7-s + (−0.809 − 0.587i)8-s + (−3.06 + 2.22i)9-s + (2.20 + 0.345i)10-s + (2.91 − 1.58i)11-s + (2.10 + 1.53i)12-s + (0.545 − 0.396i)13-s + (−0.812 + 2.50i)14-s + (−5.75 − 0.899i)15-s + (0.309 − 0.951i)16-s + (1.36 − 0.989i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.464 − 1.43i)3-s + (−0.404 + 0.293i)4-s + (0.452 − 0.891i)5-s + (0.860 − 0.625i)6-s + (0.804 + 0.584i)7-s + (−0.286 − 0.207i)8-s + (−1.02 + 0.741i)9-s + (0.698 + 0.109i)10-s + (0.877 − 0.479i)11-s + (0.608 + 0.442i)12-s + (0.151 − 0.109i)13-s + (−0.217 + 0.668i)14-s + (−1.48 − 0.232i)15-s + (0.0772 − 0.237i)16-s + (0.330 − 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25361 - 0.866509i\)
\(L(\frac12)\) \(\approx\) \(1.25361 - 0.866509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.01 + 1.99i)T \)
11 \( 1 + (-2.91 + 1.58i)T \)
good3 \( 1 + (0.805 + 2.47i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-2.12 - 1.54i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.545 + 0.396i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.36 + 0.989i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.71 + 1.97i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.178 + 0.129i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.67 + 1.22i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + 8.09T + 31T^{2} \)
37 \( 1 + (3.01 + 9.29i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + (-3.45 - 10.6i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.97 - 4.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.457 - 1.40i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.42 + 2.48i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.09 - 12.6i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 - 0.279T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 + (-0.942 - 0.684i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.72 - 7.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.11 - 2.98i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-7.94 - 5.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.94546911263050415947579541328, −9.254853471588288284773252383912, −8.692163284822881790799556335803, −7.81392154965116369212159040674, −6.95908501478499739649147552792, −5.89314013569673330010859881128, −5.51345312398810109256307591519, −4.21916705297993722154706964079, −2.18074370651862104057311476727, −0.956106160308990188605482925855, 1.80377593431617013853486582830, 3.48421644184607238958500042691, 4.16070669412298593534943076404, 5.11026205721978852994724893142, 6.10572573243958078959312373909, 7.26204197334452029861031151436, 8.693876581916550725075947652520, 9.633223782529112323808313320212, 10.29165256632845788279002017071, 10.83789342681680778597258713141

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