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Properties

Label 2-550-55.14-c1-0-2
Degree $2$
Conductor $550$
Sign $0.383 - 0.923i$
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  + (−0.587 − 0.809i)2-s + (0.363 − 0.118i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.224i)6-s + (−1.90 − 0.618i)7-s + (0.951 − 0.309i)8-s + (−2.30 + 1.67i)9-s + (0.309 + 3.30i)11-s + 0.381i·12-s + (0.726 + i)13-s + (0.618 + 1.90i)14-s + (−0.809 − 0.587i)16-s + (−0.363 + 0.5i)17-s + (2.71 + 0.881i)18-s + (1.80 + 5.56i)19-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.209 − 0.0681i)3-s + (−0.154 + 0.475i)4-s + (−0.126 − 0.0916i)6-s + (−0.718 − 0.233i)7-s + (0.336 − 0.109i)8-s + (−0.769 + 0.559i)9-s + (0.0931 + 0.995i)11-s + 0.110i·12-s + (0.201 + 0.277i)13-s + (0.165 + 0.508i)14-s + (−0.202 − 0.146i)16-s + (−0.0881 + 0.121i)17-s + (0.639 + 0.207i)18-s + (0.415 + 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636489 + 0.424637i\)
\(L(\frac12)\) \(\approx\) \(0.636489 + 0.424637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (-0.309 - 3.30i)T \)
good3 \( 1 + (-0.363 + 0.118i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.90 + 0.618i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.726 - i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.80 - 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.52 + 1.14i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.73 - 5.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.56iT - 43T^{2} \)
47 \( 1 + (-6.15 + 2i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.898 + 1.23i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.66 + 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2 - 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.87 + 3.20i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.8 + 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.48 - 7.54i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.09T + 89T^{2} \)
97 \( 1 + (4.20 + 5.78i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.84643435295484378914301868834, −10.04594632522745601371659503965, −9.342096748713698676965817630428, −8.394426749740750217178631536058, −7.54952543684227449714657932221, −6.56449931839058106763494176873, −5.32130288730316766244777321318, −4.02525682824393530880320784440, −2.97868518046444213905754646760, −1.71783306167050437423867911922, 0.48651015681231502118986217009, 2.71030161058297376455572280258, 3.75681849931537113445423730136, 5.37141122675025620363821270382, 6.08016989366671793255653858512, 6.97274519470814640944693833690, 8.066032838997551502482035045930, 9.022353875623730424139478431284, 9.318063847429504680026320172973, 10.55176040343115216614129130895

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