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Properties

Label 2-550-275.16-c1-0-24
Degree $2$
Conductor $550$
Sign $-0.997 - 0.0642i$
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 + (−0.391 + 1.20i)3-s + (−0.809 − 0.587i)4-s + (−1.60 − 1.56i)5-s + (1.02 + 0.744i)6-s + (−1.25 + 0.913i)7-s + (−0.809 + 0.587i)8-s + (1.12 + 0.820i)9-s + (−1.97 + 1.04i)10-s + (2.51 − 2.16i)11-s + (1.02 − 0.744i)12-s + (−2.87 − 2.08i)13-s + (0.480 + 1.47i)14-s + (2.50 − 1.31i)15-s + (0.309 + 0.951i)16-s + (−6.33 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.226 + 0.695i)3-s + (−0.404 − 0.293i)4-s + (−0.715 − 0.698i)5-s + (0.418 + 0.303i)6-s + (−0.475 + 0.345i)7-s + (−0.286 + 0.207i)8-s + (0.376 + 0.273i)9-s + (−0.625 + 0.328i)10-s + (0.758 − 0.652i)11-s + (0.295 − 0.214i)12-s + (−0.796 − 0.578i)13-s + (0.128 + 0.394i)14-s + (0.647 − 0.340i)15-s + (0.0772 + 0.237i)16-s + (−1.53 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $-0.997 - 0.0642i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ -0.997 - 0.0642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0102273 + 0.317876i\)
\(L(\frac12)\) \(\approx\) \(0.0102273 + 0.317876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 1.56i)T \)
11 \( 1 + (-2.51 + 2.16i)T \)
good3 \( 1 + (0.391 - 1.20i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.25 - 0.913i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.87 + 2.08i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.33 + 4.59i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.06 - 4.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.09 + 2.97i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.29 + 1.66i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 - 0.336T + 31T^{2} \)
37 \( 1 + (-1.68 + 5.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 - 5.91T + 43T^{2} \)
47 \( 1 + (-0.545 + 1.67i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.29 + 3.11i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0523 + 0.161i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.17 + 3.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.86 + 5.72i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + (-4.98 + 3.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.0 - 7.33i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.72 + 1.97i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.81 + 1.31i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.46517784952771986704801082307, −9.507524520406809765195760831850, −8.872123558181863447909970126215, −7.87563458653731603938966564500, −6.51883650313323743362724604927, −5.34639961291829671658566731008, −4.39962910540601101381687438697, −3.78997743335321695633782451175, −2.24627775646844132641753510154, −0.16629268681813672781761318446, 2.13840955495451817634015217052, 3.92517734323489329002917420212, 4.42558124393389792309487600205, 6.28768469736211878146563019442, 6.78323935048065170447183920039, 7.22457646519160299505376899373, 8.363539956954561293598059313944, 9.371741342917571942796157121326, 10.36696224334684713404388082977, 11.42637994613147026342801471590

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