Properties

Label 2-546-21.17-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.330 + 0.943i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (−0.866 + 0.5i)2-s + (−0.526 + 1.65i)3-s + (0.499 − 0.866i)4-s + (1.35 + 2.34i)5-s + (−0.369 − 1.69i)6-s + (−2.61 − 0.408i)7-s + 0.999i·8-s + (−2.44 − 1.73i)9-s + (−2.34 − 1.35i)10-s + (−4.42 − 2.55i)11-s + (1.16 + 1.28i)12-s i·13-s + (2.46 − 0.953i)14-s + (−4.58 + 0.998i)15-s + (−0.5 − 0.866i)16-s + (2.20 − 3.82i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.304 + 0.952i)3-s + (0.249 − 0.433i)4-s + (0.605 + 1.04i)5-s + (−0.150 − 0.690i)6-s + (−0.988 − 0.154i)7-s + 0.353i·8-s + (−0.815 − 0.579i)9-s + (−0.741 − 0.428i)10-s + (−1.33 − 0.770i)11-s + (0.336 + 0.369i)12-s − 0.277i·13-s + (0.659 − 0.254i)14-s + (−1.18 + 0.257i)15-s + (−0.125 − 0.216i)16-s + (0.535 − 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.330 + 0.943i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0482486 - 0.0680335i\)
\(L(\frac12)\) \(\approx\) \(0.0482486 - 0.0680335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.526 - 1.65i)T \)
7 \( 1 + (2.61 + 0.408i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-1.35 - 2.34i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.42 + 2.55i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.20 + 3.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.03 - 1.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 + (-0.214 - 0.123i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.86 + 3.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.85T + 41T^{2} \)
43 \( 1 - 6.76T + 43T^{2} \)
47 \( 1 + (5.96 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.86 + 5.69i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.40 - 9.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.805 - 0.465i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.327 - 0.567i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 + (5.12 + 2.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.694 + 1.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.94T + 83T^{2} \)
89 \( 1 + (5.85 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−10.80864872820554066330091930672, −10.44653658391715833695516800360, −9.865918346232594713059492674326, −8.992648849264187853733215391668, −7.87453297644781523774545917857, −6.78779684568812172956340122826, −5.93945432165055875745729662978, −5.26968943761153526696798980404, −3.49191565450677793142890964744, −2.68478040971270309424859214468, 0.05644258484871780954746566789, 1.71720669922769286099561346487, 2.70225207843370941299612764966, 4.54772825697703483833007795886, 5.79538398089633795026849264689, 6.47548446087502045180218490927, 7.74426038981866665166964312933, 8.324794310715451824096505283631, 9.396108156704103611112373434840, 10.07894691255791527737353760401

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