Properties

Label 2-546-273.257-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.0957 + 0.995i$
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 + (−1.40 + 1.00i)3-s + 4-s + (−1.41 − 0.815i)5-s + (1.40 − 1.00i)6-s + (2.62 − 0.292i)7-s − 8-s + (0.974 − 2.83i)9-s + (1.41 + 0.815i)10-s + (−1.03 + 1.79i)11-s + (−1.40 + 1.00i)12-s + (−3.37 + 1.27i)13-s + (−2.62 + 0.292i)14-s + (2.81 − 0.272i)15-s + 16-s − 3.05·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.813 + 0.581i)3-s + 0.5·4-s + (−0.631 − 0.364i)5-s + (0.575 − 0.410i)6-s + (0.993 − 0.110i)7-s − 0.353·8-s + (0.324 − 0.945i)9-s + (0.446 + 0.257i)10-s + (−0.312 + 0.540i)11-s + (−0.406 + 0.290i)12-s + (−0.935 + 0.352i)13-s + (−0.702 + 0.0782i)14-s + (0.726 − 0.0702i)15-s + 0.250·16-s − 0.740·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0957 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0957 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0957 + 0.995i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0957 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267278 - 0.294215i\)
\(L(\frac12)\) \(\approx\) \(0.267278 - 0.294215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 1.00i)T \)
7 \( 1 + (-2.62 + 0.292i)T \)
13 \( 1 + (3.37 - 1.27i)T \)
good5 \( 1 + (1.41 + 0.815i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.03 - 1.79i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + (-0.662 - 1.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.10iT - 23T^{2} \)
29 \( 1 + (-2.79 + 1.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.28 + 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.93iT - 37T^{2} \)
41 \( 1 + (-7.41 + 4.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.69 + 4.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.714 + 0.412i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.25 - 5.34i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.5iT - 59T^{2} \)
61 \( 1 + (13.4 - 7.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.83 + 1.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.59 + 7.95i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.43 - 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 - 5.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.85iT - 83T^{2} \)
89 \( 1 + 1.07iT - 89T^{2} \)
97 \( 1 + (4.58 - 7.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.71844508823443674889653269343, −9.715001107182721673832682495361, −8.914003774236895744305150703051, −7.87726928206765292914516276681, −7.15815248942553798096150670231, −5.95322480780810031931147759138, −4.73900934115590190605067115739, −4.20175528564395477788483446014, −2.19048997002107188776989152016, −0.33095873928119851116167805018, 1.42886283921132908879227275393, 2.88745037305452707902352924731, 4.66574099472069717161319754003, 5.56611664625444117654918830301, 6.74091313272051250062486668223, 7.62314771901487888085978980512, 8.027020104353126008239402653795, 9.237980862196549840332192047968, 10.42839837377127770096468153021, 11.15982509928992465813360148983

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