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Properties

Label 2-550-55.52-c1-0-6
Degree $2$
Conductor $550$
Sign $0.568 + 0.822i$
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  + (−0.156 + 0.987i)2-s + (−2.86 − 1.45i)3-s + (−0.951 − 0.309i)4-s + (1.88 − 2.59i)6-s + (0.692 + 1.35i)7-s + (0.453 − 0.891i)8-s + (4.29 + 5.91i)9-s + (−2.70 + 1.91i)11-s + (2.27 + 2.27i)12-s + (−1.04 − 0.165i)13-s + (−1.44 + 0.471i)14-s + (0.809 + 0.587i)16-s + (2.19 − 0.347i)17-s + (−6.51 + 3.31i)18-s + (−1.91 − 5.89i)19-s + ⋯
L(s)  = 1  + (−0.110 + 0.698i)2-s + (−1.65 − 0.841i)3-s + (−0.475 − 0.154i)4-s + (0.770 − 1.06i)6-s + (0.261 + 0.513i)7-s + (0.160 − 0.315i)8-s + (1.43 + 1.97i)9-s + (−0.815 + 0.578i)11-s + (0.655 + 0.655i)12-s + (−0.289 − 0.0459i)13-s + (−0.387 + 0.125i)14-s + (0.202 + 0.146i)16-s + (0.532 − 0.0843i)17-s + (−1.53 + 0.782i)18-s + (−0.439 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.568 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472159 - 0.247770i\)
\(L(\frac12)\) \(\approx\) \(0.472159 - 0.247770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 - 0.987i)T \)
5 \( 1 \)
11 \( 1 + (2.70 - 1.91i)T \)
good3 \( 1 + (2.86 + 1.45i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-0.692 - 1.35i)T + (-4.11 + 5.66i)T^{2} \)
13 \( 1 + (1.04 + 0.165i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.19 + 0.347i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.05 + 3.05i)T - 23iT^{2} \)
29 \( 1 + (-1.39 + 4.30i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.32 + 1.69i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.24 - 0.631i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-2.38 + 0.774i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (6.40 + 6.40i)T + 43iT^{2} \)
47 \( 1 + (-2.03 + 3.99i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (0.594 - 3.75i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-4.27 - 1.38i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.48 + 4.79i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (5.40 + 5.40i)T + 67iT^{2} \)
71 \( 1 + (7.19 + 5.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-10.1 + 5.17i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (4.69 - 3.40i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.200 - 1.26i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 + (6.92 + 1.09i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.71111200066158910704089888580, −9.983366987769231914223370041370, −8.648037579123344495804605874087, −7.60775323928613313186085906880, −6.96190852711155025746955514558, −6.12367837106725070632726034132, −5.19227317293486294358921569426, −4.69720482348279127004380132735, −2.24995317523736227293759851276, −0.47392733966921224581420842577, 1.14079428496128368764235087222, 3.33146406400879447505902248495, 4.37592607475896992128358148180, 5.24504009211110473549136970155, 5.99207349850656498794685808841, 7.25170011836935281093855677527, 8.426905060753745196655435710591, 9.728923492817438836433038286050, 10.27776280960866697252475220595, 10.88813549912848848968745449740

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