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Properties

Label 2-549-61.48-c1-0-12
Degree $2$
Conductor $549$
Sign $-0.473 - 0.881i$
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  + (2.40 + 1.38i)2-s + (2.85 + 4.95i)4-s + (−0.766 + 1.32i)5-s + (−1.02 − 0.590i)7-s + 10.3i·8-s + (−3.68 + 2.12i)10-s − 2.65i·11-s + (0.803 − 1.39i)13-s + (−1.63 − 2.83i)14-s + (−8.62 + 14.9i)16-s + (2.95 − 1.70i)17-s + (−0.255 − 0.443i)19-s − 8.76·20-s + (3.69 − 6.39i)22-s − 2.79i·23-s + ⋯
L(s)  = 1  + (1.70 + 0.982i)2-s + (1.42 + 2.47i)4-s + (−0.342 + 0.593i)5-s + (−0.386 − 0.223i)7-s + 3.64i·8-s + (−1.16 + 0.673i)10-s − 0.801i·11-s + (0.222 − 0.385i)13-s + (−0.438 − 0.758i)14-s + (−2.15 + 3.73i)16-s + (0.715 − 0.413i)17-s + (−0.0587 − 0.101i)19-s − 1.95·20-s + (0.787 − 1.36i)22-s − 0.582i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(549\)    =    \(3^{2} \cdot 61\)
Sign: $-0.473 - 0.881i$
Analytic conductor: \(4.38378\)
Root analytic conductor: \(2.09374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{549} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 549,\ (\ :1/2),\ -0.473 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76970 + 2.95898i\)
\(L(\frac12)\) \(\approx\) \(1.76970 + 2.95898i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.39 + 6.45i)T \)
good2 \( 1 + (-2.40 - 1.38i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.766 - 1.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.02 + 0.590i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.65iT - 11T^{2} \)
13 \( 1 + (-0.803 + 1.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 1.70i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.255 + 0.443i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.79iT - 23T^{2} \)
29 \( 1 + (-1.92 + 1.11i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.66 - 1.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.01iT - 37T^{2} \)
41 \( 1 - 7.87T + 41T^{2} \)
43 \( 1 + (-8.56 - 4.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.952 + 1.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + (10.7 + 6.18i)T + (29.5 + 51.0i)T^{2} \)
67 \( 1 + (8.09 + 4.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.5 - 6.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.48 + 4.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.21 - 1.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.26 + 9.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.22iT - 89T^{2} \)
97 \( 1 + (2.17 + 3.77i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.26809854120910869556509567418, −10.69615263959862723293330056358, −8.997253354382860671952403617037, −7.85204409380541068682659742088, −7.29959177028640869035598390237, −6.30562521340335767701759371817, −5.67147422067437836253494059273, −4.51682404514064109566785771947, −3.46953972296869854123908963940, −2.83058409417846976266845021182, 1.37859211515613667541679806455, 2.69089338905016882451558357035, 3.85967441948368138461919327675, 4.58517591311418801984700837126, 5.56529218114194160620596091161, 6.41017487800407537764029838816, 7.52387458110037801807404888709, 9.109996006908598964245137059770, 9.967342961734618453343537013711, 10.76543594656372713414609794915

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