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Properties

Label 2-549-61.14-c1-0-1
Degree $2$
Conductor $549$
Sign $-0.614 - 0.788i$
Analytic cond. $4.38378$
Root an. cond. $2.09374$
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  + (−2.15 + 1.24i)2-s + (2.08 − 3.60i)4-s + (−1.17 − 2.03i)5-s + (−2.32 + 1.33i)7-s + 5.38i·8-s + (5.06 + 2.92i)10-s − 0.681i·11-s + (−2.09 − 3.62i)13-s + (3.32 − 5.76i)14-s + (−2.51 − 4.35i)16-s + (0.990 + 0.572i)17-s + (−1.14 + 1.98i)19-s − 9.80·20-s + (0.846 + 1.46i)22-s + 6.21i·23-s + ⋯
L(s)  = 1  + (−1.52 + 0.878i)2-s + (1.04 − 1.80i)4-s + (−0.526 − 0.911i)5-s + (−0.877 + 0.506i)7-s + 1.90i·8-s + (1.60 + 0.924i)10-s − 0.205i·11-s + (−0.579 − 1.00i)13-s + (0.889 − 1.54i)14-s + (−0.629 − 1.08i)16-s + (0.240 + 0.138i)17-s + (−0.262 + 0.454i)19-s − 2.19·20-s + (0.180 + 0.312i)22-s + 1.29i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(549\)    =    \(3^{2} \cdot 61\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(4.38378\)
Root analytic conductor: \(2.09374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{549} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 549,\ (\ :1/2),\ -0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131638 + 0.269512i\)
\(L(\frac12)\) \(\approx\) \(0.131638 + 0.269512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 + (7.68 - 1.40i)T \)
good2 \( 1 + (2.15 - 1.24i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.17 + 2.03i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.32 - 1.33i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.681iT - 11T^{2} \)
13 \( 1 + (2.09 + 3.62i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.990 - 0.572i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.14 - 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.21iT - 23T^{2} \)
29 \( 1 + (-2.79 - 1.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.38 - 4.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.43iT - 37T^{2} \)
41 \( 1 - 0.405T + 41T^{2} \)
43 \( 1 + (1.28 - 0.739i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.68iT - 53T^{2} \)
59 \( 1 + (1.80 - 1.03i)T + (29.5 - 51.0i)T^{2} \)
67 \( 1 + (7.07 - 4.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 1.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.94 + 5.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.50 + 4.90i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.67 - 9.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.38iT - 89T^{2} \)
97 \( 1 + (3.93 - 6.81i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66550417899789064893262712195, −9.857816488876560773164537352304, −9.276372993766841256328654615509, −8.237562904980249291859374718023, −7.981284838060360066859902407601, −6.74741056885752345106932187651, −5.91603677473761126004647992929, −4.90932910944526975962289028078, −3.10214848810926481042015203297, −1.14493956577252543785481746882, 0.32737845162880571762124314939, 2.27141894930770497669425118789, 3.17763404201023045251615381648, 4.33112819300378774840973627119, 6.53219761247773704464425042170, 7.07568008496707849210364655508, 7.917411380188760062388663631894, 8.969609768282247257425981912268, 9.749879336045626975548564250079, 10.39506241409949301156797900638

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