Properties

Label 2-546-21.20-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.218 - 0.975i$
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  i·2-s + (−1.61 + 0.618i)3-s − 4-s − 1.23·5-s + (0.618 + 1.61i)6-s + (0.381 − 2.61i)7-s + i·8-s + (2.23 − 2.00i)9-s + 1.23i·10-s + (1.61 − 0.618i)12-s + i·13-s + (−2.61 − 0.381i)14-s + (2.00 − 0.763i)15-s + 16-s − 6.47·17-s + (−2.00 − 2.23i)18-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.934 + 0.356i)3-s − 0.5·4-s − 0.552·5-s + (0.252 + 0.660i)6-s + (0.144 − 0.989i)7-s + 0.353i·8-s + (0.745 − 0.666i)9-s + 0.390i·10-s + (0.467 − 0.178i)12-s + 0.277i·13-s + (−0.699 − 0.102i)14-s + (0.516 − 0.197i)15-s + 0.250·16-s − 1.56·17-s + (−0.471 − 0.527i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137154 + 0.171210i\)
\(L(\frac12)\) \(\approx\) \(0.137154 + 0.171210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.61 - 0.618i)T \)
7 \( 1 + (-0.381 + 2.61i)T \)
13 \( 1 - iT \)
good5 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 1.23iT - 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 + 0.763T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 4.47iT - 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 5.52iT - 71T^{2} \)
73 \( 1 - 1.23iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + 5.23iT - 97T^{2} \)
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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

−10.94293650779809128354135629313, −10.50969770103306145983237414173, −9.595382528803111981652385344534, −8.542261255176508036876940730888, −7.38004250907892275496881373515, −6.50943660639169949410297023996, −5.21309367697032257563326574976, −4.24853661178930998788238765393, −3.60470593542013135407300783335, −1.54522890431399930719550629383, 0.14492481619184129478389763290, 2.32638493614287781146696039331, 4.23861201714527554878012662128, 5.07030838922238432529418584516, 6.04893702698268726148281776814, 6.78850499279001052125873378114, 7.73992541799218099319462437319, 8.616643449162671273166745190444, 9.511357783977438458691403276512, 10.75042178671790385824993338845

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