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Properties

Label 2-546-273.272-c1-0-28
Degree $2$
Conductor $546$
Sign $0.685 + 0.728i$
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  + 2-s + (0.396 − 1.68i)3-s + 4-s + 2.37i·5-s + (0.396 − 1.68i)6-s + (0.792 − 2.52i)7-s + 8-s + (−2.68 − 1.33i)9-s + 2.37i·10-s + 5.74·11-s + (0.396 − 1.68i)12-s + (−3.46 − i)13-s + (0.792 − 2.52i)14-s + (4 + 0.939i)15-s + 16-s − 2.52·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.228 − 0.973i)3-s + 0.5·4-s + 1.06i·5-s + (0.161 − 0.688i)6-s + (0.299 − 0.954i)7-s + 0.353·8-s + (−0.895 − 0.445i)9-s + 0.750i·10-s + 1.73·11-s + (0.114 − 0.486i)12-s + (−0.960 − 0.277i)13-s + (0.211 − 0.674i)14-s + (1.03 + 0.242i)15-s + 0.250·16-s − 0.612·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27915 - 0.985061i\)
\(L(\frac12)\) \(\approx\) \(2.27915 - 0.985061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.396 + 1.68i)T \)
7 \( 1 + (-0.792 + 2.52i)T \)
13 \( 1 + (3.46 + i)T \)
good5 \( 1 - 2.37iT - 5T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 - 7.57T + 19T^{2} \)
23 \( 1 + 6.78iT - 23T^{2} \)
29 \( 1 - 7.57iT - 29T^{2} \)
31 \( 1 + 5.84T + 31T^{2} \)
37 \( 1 - 4.90iT - 37T^{2} \)
41 \( 1 - 2.62iT - 41T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 + 4.62iT - 47T^{2} \)
53 \( 1 - 3.16iT - 53T^{2} \)
59 \( 1 - 8.74iT - 59T^{2} \)
61 \( 1 + 3.74iT - 61T^{2} \)
67 \( 1 - 2.67iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 3.61T + 73T^{2} \)
79 \( 1 + 9.37T + 79T^{2} \)
83 \( 1 - 4.74iT - 83T^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + 7.42T + 97T^{2} \)
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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

−10.97709274171450805468954010152, −9.973757669124735798167233912716, −8.805163147234686117647759669919, −7.52716362163611091759966349044, −6.92390324824805067431763907155, −6.51055142053595585407815700676, −5.06045239346437786960517159697, −3.72681711696237823516420305812, −2.83984808060290603046460499705, −1.38180897512599059992427327185, 1.84292895993510538927001192635, 3.36138237782426975045777791441, 4.37973124516603654427053034982, 5.16116917247425565542594076053, 5.87341830582384439038235395713, 7.31110668599253132701773203293, 8.496458280633405105915488532786, 9.456179020237659057703593739982, 9.543663759456154561807352476266, 11.38628127637904435169483329110

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