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Properties

Label 2-546-273.200-c1-0-2
Degree $2$
Conductor $546$
Sign $0.0167 - 0.999i$
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  + (0.258 − 0.965i)2-s + (−1.51 + 0.831i)3-s + (−0.866 − 0.499i)4-s + (−0.533 + 1.98i)5-s + (0.410 + 1.68i)6-s + (1.42 − 2.22i)7-s + (−0.707 + 0.707i)8-s + (1.61 − 2.52i)9-s + (1.78 + 1.02i)10-s + (0.922 + 3.44i)11-s + (1.73 + 0.0392i)12-s + (−3.26 − 1.53i)13-s + (−1.78 − 1.95i)14-s + (−0.844 − 3.46i)15-s + (0.500 + 0.866i)16-s + (−3.54 + 6.14i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.877 + 0.480i)3-s + (−0.433 − 0.249i)4-s + (−0.238 + 0.889i)5-s + (0.167 + 0.686i)6-s + (0.539 − 0.841i)7-s + (−0.249 + 0.249i)8-s + (0.538 − 0.842i)9-s + (0.564 + 0.325i)10-s + (0.278 + 1.03i)11-s + (0.499 + 0.0113i)12-s + (−0.904 − 0.426i)13-s + (−0.476 − 0.522i)14-s + (−0.218 − 0.894i)15-s + (0.125 + 0.216i)16-s + (−0.860 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0167 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0167 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491399 + 0.483214i\)
\(L(\frac12)\) \(\approx\) \(0.491399 + 0.483214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.51 - 0.831i)T \)
7 \( 1 + (-1.42 + 2.22i)T \)
13 \( 1 + (3.26 + 1.53i)T \)
good5 \( 1 + (0.533 - 1.98i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.922 - 3.44i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.54 - 6.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.57 + 1.49i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.12 - 5.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.340iT - 29T^{2} \)
31 \( 1 + (-2.22 - 8.31i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.01 - 7.50i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.27 + 5.27i)T + 41iT^{2} \)
43 \( 1 - 0.428iT - 43T^{2} \)
47 \( 1 + (1.50 + 0.403i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.74 + 3.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.522 + 1.94i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.887 + 1.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.14 - 11.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.49 - 1.49i)T + 71iT^{2} \)
73 \( 1 + (-4.28 + 1.14i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.48 + 7.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.13 - 8.13i)T + 83iT^{2} \)
89 \( 1 + (2.24 + 0.601i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.9 + 11.9i)T - 97iT^{2} \)
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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

−10.79589056049751980612405579438, −10.52264213974225873402662139179, −9.746472104572009231633341217116, −8.476827766267982584845666883866, −7.10031348541371547730436364150, −6.60172223593161354712510363609, −5.07493838851256000394529869926, −4.39180717092370751728888949093, −3.40177490189908236422291068184, −1.71429333810278053306307894216, 0.41804544132327215009574717284, 2.34217823848867128131256808766, 4.51848618611855095819577674907, 4.93368881834518019754886091847, 6.00646376430565563570311775754, 6.76599647735599069750329763165, 7.894357707837692535059153429075, 8.655607046027244257915944306382, 9.384250119077310154857703896250, 10.88162198992317885011312965585

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