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Properties

Label 2-550-11.4-c1-0-16
Degree $2$
Conductor $550$
Sign $0.314 + 0.949i$
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  + (0.809 + 0.587i)2-s + (0.5 − 1.53i)3-s + (0.309 + 0.951i)4-s + (1.30 − 0.951i)6-s + (−1.54 − 4.76i)7-s + (−0.309 + 0.951i)8-s + (0.309 + 0.224i)9-s + (−0.969 − 3.17i)11-s + 1.61·12-s + (−1.88 − 1.37i)13-s + (1.54 − 4.76i)14-s + (−0.809 + 0.587i)16-s + (−0.0494 + 0.0359i)17-s + (0.118 + 0.363i)18-s + (−0.636 + 1.96i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.288 − 0.888i)3-s + (0.154 + 0.475i)4-s + (0.534 − 0.388i)6-s + (−0.585 − 1.80i)7-s + (−0.109 + 0.336i)8-s + (0.103 + 0.0748i)9-s + (−0.292 − 0.956i)11-s + 0.467·12-s + (−0.523 − 0.380i)13-s + (0.414 − 1.27i)14-s + (−0.202 + 0.146i)16-s + (−0.0119 + 0.00871i)17-s + (0.0278 + 0.0856i)18-s + (−0.146 + 0.449i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57326 - 1.13556i\)
\(L(\frac12)\) \(\approx\) \(1.57326 - 1.13556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
11 \( 1 + (0.969 + 3.17i)T \)
good3 \( 1 + (-0.5 + 1.53i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.54 + 4.76i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.88 + 1.37i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0494 - 0.0359i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.636 - 1.96i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.91T + 23T^{2} \)
29 \( 1 + (1.27 + 3.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.18 - 5.21i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.40 - 4.33i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.41 - 4.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.61T + 43T^{2} \)
47 \( 1 + (-2.74 + 8.44i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.79 + 1.30i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.599 - 1.84i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.84 + 5.69i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.49T + 67T^{2} \)
71 \( 1 + (7.13 - 5.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.66 - 5.13i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.38 - 3.91i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.94 - 6.49i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + (-5.90 - 4.29i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.56834284452477203786378624759, −9.939191486250626601643903543221, −8.388877105783864736126789577343, −7.72793886394795568918189275637, −6.97038624863716372117627512590, −6.34063606627013631174013002364, −4.95540834343492030915095280659, −3.84941254674470441719693048215, −2.80727263070080031999593667181, −0.936314851836682536098054823893, 2.24859493789593086279761762002, 3.06143545473026723043816999094, 4.38359058292658591377835011541, 5.12326141747653869523248597668, 6.14913280308214867558950280459, 7.22445508660060874374092207882, 8.774381158747919114706943936294, 9.381439993148592387028768775824, 9.928847726376719834180531378970, 10.96914353980788697193766041225

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