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Properties

Label 2-550-55.32-c1-0-2
Degree $2$
Conductor $550$
Sign $-0.622 - 0.782i$
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  + (−0.707 + 0.707i)2-s + (1 − i)3-s − 1.00i·4-s + 1.41i·6-s + (−2.82 + 2.82i)7-s + (0.707 + 0.707i)8-s + i·9-s + (−3 + 1.41i)11-s + (−1.00 − 1.00i)12-s + (1.41 + 1.41i)13-s − 4.00i·14-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−0.707 − 0.707i)18-s − 8.48·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.577 − 0.577i)3-s − 0.500i·4-s + 0.577i·6-s + (−1.06 + 1.06i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.904 + 0.426i)11-s + (−0.288 − 0.288i)12-s + (0.392 + 0.392i)13-s − 1.06i·14-s − 0.250·16-s + (0.342 − 0.342i)17-s + (−0.166 − 0.166i)18-s − 1.94·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ -0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320189 + 0.664145i\)
\(L(\frac12)\) \(\approx\) \(0.320189 + 0.664145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
11 \( 1 + (3 - 1.41i)T \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (2.82 - 2.82i)T - 7iT^{2} \)
13 \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 + 8.48T + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + (-2.82 - 2.82i)T + 43iT^{2} \)
47 \( 1 + (-1 - i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + 14.1iT - 61T^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (7.07 + 7.07i)T + 73iT^{2} \)
79 \( 1 - 2.82T + 79T^{2} \)
83 \( 1 + (5.65 + 5.65i)T + 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-7 - 7i)T + 97iT^{2} \)
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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

−10.89066481494742265692251709833, −10.00257622073694686361141484891, −9.167728997328724242210202507910, −8.368351734886156581992270333784, −7.71082394977352466545176247777, −6.59550377988429378278499523272, −5.93762767602204364548489402459, −4.67770975293161068727991561326, −2.91876818002377319181911031941, −2.00520936223698616739704904486, 0.43893739910340033527428080513, 2.57277604801762574354164224214, 3.60424085191885521993802342908, 4.26049608900297004599547513572, 6.02925844404980545312955001098, 6.94181453631629259622781549775, 8.183326393388477196396370272273, 8.715760402518198050695357533012, 9.835501670182246546319501293796, 10.38767582418950655769071600359

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