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Properties

Label 2-550-25.4-c1-0-12
Degree $2$
Conductor $550$
Sign $0.856 + 0.515i$
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  + (−0.951 − 0.309i)2-s + (−0.260 + 0.357i)3-s + (0.809 + 0.587i)4-s + (−0.166 − 2.22i)5-s + (0.357 − 0.260i)6-s + 0.273i·7-s + (−0.587 − 0.809i)8-s + (0.866 + 2.66i)9-s + (−0.530 + 2.17i)10-s + (−0.309 + 0.951i)11-s + (−0.420 + 0.136i)12-s + (2.5 − 0.812i)13-s + (0.0845 − 0.260i)14-s + (0.841 + 0.520i)15-s + (0.309 + 0.951i)16-s + (0.623 + 0.857i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (−0.150 + 0.206i)3-s + (0.404 + 0.293i)4-s + (−0.0746 − 0.997i)5-s + (0.146 − 0.106i)6-s + 0.103i·7-s + (−0.207 − 0.286i)8-s + (0.288 + 0.888i)9-s + (−0.167 + 0.686i)10-s + (−0.0931 + 0.286i)11-s + (−0.121 + 0.0394i)12-s + (0.693 − 0.225i)13-s + (0.0225 − 0.0695i)14-s + (0.217 + 0.134i)15-s + (0.0772 + 0.237i)16-s + (0.151 + 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.856 + 0.515i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01066 - 0.280614i\)
\(L(\frac12)\) \(\approx\) \(1.01066 - 0.280614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (0.166 + 2.22i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.260 - 0.357i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 - 0.273iT - 7T^{2} \)
13 \( 1 + (-2.5 + 0.812i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.623 - 0.857i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.55 + 4.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.09 + 1.32i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.40 - 1.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.78 + 4.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.2 + 3.33i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.55 - 7.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.85iT - 43T^{2} \)
47 \( 1 + (-2.81 + 3.87i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.20 + 5.78i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.546 + 1.68i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.33 - 13.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.21 + 11.3i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.48 + 3.98i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.51 + 0.816i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-13.5 - 9.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.49 - 7.56i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.44 + 13.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.88 - 3.96i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.60497279651562634765028527716, −9.776698692215538036942738954041, −9.051555161196451160080566624915, −8.069395197321511694237649335316, −7.51218491997564326477556612319, −6.06759407319958901953489366331, −5.05771823204959195856367884927, −4.08110903172708504086679736960, −2.47213227731800486126164618825, −0.995593786623640741540248764170, 1.19014390076396306773597844741, 2.91522726636804120375780931004, 3.96843081632990997603459638482, 5.80896617146977306019608362546, 6.35372529023328817928161148244, 7.36864439504560254159891278960, 8.003305876264956526708993868565, 9.244576818818003561354046983729, 9.961988054844961963751182535584, 10.71575879205061782264026319127

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