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Properties

Label 2-546-39.2-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.682 - 0.731i$
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 + (−0.413 − 1.68i)3-s + (−0.866 − 0.499i)4-s + (−2.27 − 2.27i)5-s + (1.73 + 0.0363i)6-s + (−0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−2.65 + 1.38i)9-s + (2.78 − 1.60i)10-s + (−1.90 − 0.511i)11-s + (−0.483 + 1.66i)12-s + (1.84 + 3.10i)13-s − i·14-s + (−2.88 + 4.76i)15-s + (0.500 + 0.866i)16-s + (0.403 − 0.698i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.238 − 0.971i)3-s + (−0.433 − 0.249i)4-s + (−1.01 − 1.01i)5-s + (0.706 + 0.0148i)6-s + (−0.365 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (−0.886 + 0.463i)9-s + (0.880 − 0.508i)10-s + (−0.575 − 0.154i)11-s + (−0.139 + 0.480i)12-s + (0.510 + 0.859i)13-s − 0.267i·14-s + (−0.744 + 1.22i)15-s + (0.125 + 0.216i)16-s + (0.0978 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.682 - 0.731i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0647189 + 0.148932i\)
\(L(\frac12)\) \(\approx\) \(0.0647189 + 0.148932i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.413 + 1.68i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (-1.84 - 3.10i)T \)
good5 \( 1 + (2.27 + 2.27i)T + 5iT^{2} \)
11 \( 1 + (1.90 + 0.511i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.403 + 0.698i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.40 - 5.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.417 - 0.722i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.40 - 3.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 2.93i)T - 31iT^{2} \)
37 \( 1 + (1.64 - 6.14i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.963 - 3.59i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.1 + 5.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.93 - 3.93i)T - 47iT^{2} \)
53 \( 1 + 8.62iT - 53T^{2} \)
59 \( 1 + (-0.382 - 1.42i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.166 + 0.288i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.60 + 1.50i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (12.3 - 3.31i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.04 - 2.04i)T + 73iT^{2} \)
79 \( 1 + 1.02T + 79T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 + (2.34 + 0.629i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.61 - 13.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.49473807798150036915004100674, −10.13153607149874301466109876758, −8.942501505309927916267950345253, −8.274607022939609385663244271871, −7.65352066040338056316539912109, −6.70348501086890505309839231552, −5.72407200504143226807483021836, −4.79489326317346246306724905863, −3.49806142362189496582922717132, −1.45964643195735951605878832068, 0.10701053997064930657975597902, 2.86742900179471785574284375001, 3.45288787903076257573519154050, 4.48951671886875213792751057297, 5.65804642598676549906498899049, 6.95616473468718164139975366469, 7.937104818149795316254379195512, 8.874847285383842649001197656395, 9.929451262356579111271476301389, 10.58560512013521640079571354151

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