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Properties

Label 2-546-39.32-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.896 + 0.442i$
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  + (0.965 − 0.258i)2-s + (0.162 − 1.72i)3-s + (0.866 − 0.499i)4-s + (−2.61 − 2.61i)5-s + (−0.289 − 1.70i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−2.94 − 0.559i)9-s + (−3.19 − 1.84i)10-s + (1.56 + 5.82i)11-s + (−0.721 − 1.57i)12-s + (1.58 − 3.23i)13-s − i·14-s + (−4.92 + 4.07i)15-s + (0.500 − 0.866i)16-s + (−1.62 − 2.81i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.0936 − 0.995i)3-s + (0.433 − 0.249i)4-s + (−1.16 − 1.16i)5-s + (−0.118 − 0.697i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.982 − 0.186i)9-s + (−1.01 − 0.583i)10-s + (0.470 + 1.75i)11-s + (−0.208 − 0.454i)12-s + (0.440 − 0.897i)13-s − 0.267i·14-s + (−1.27 + 1.05i)15-s + (0.125 − 0.216i)16-s + (−0.394 − 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365307 - 1.56635i\)
\(L(\frac12)\) \(\approx\) \(0.365307 - 1.56635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.162 + 1.72i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-1.58 + 3.23i)T \)
good5 \( 1 + (2.61 + 2.61i)T + 5iT^{2} \)
11 \( 1 + (-1.56 - 5.82i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.15 + 0.577i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.52 + 2.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.11 + 3.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.91 - 4.91i)T - 31iT^{2} \)
37 \( 1 + (-7.89 + 2.11i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.19 + 1.12i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.69 + 3.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.62 + 6.62i)T - 47iT^{2} \)
53 \( 1 + 4.59iT - 53T^{2} \)
59 \( 1 + (-8.67 - 2.32i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.42 - 5.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.149 - 0.558i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.31 + 8.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.971 + 0.971i)T + 73iT^{2} \)
79 \( 1 + 4.53T + 79T^{2} \)
83 \( 1 + (-1.86 - 1.86i)T + 83iT^{2} \)
89 \( 1 + (1.68 + 6.30i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.38 + 0.908i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.79898584275968511689328858367, −9.374219054513274942533429095804, −8.494367867000306182534017074229, −7.46578466886604502926192249747, −7.08118879076257479210839376412, −5.63944420290859144480629232806, −4.60411738558613659238683723678, −3.81850219649864218269179634340, −2.18850250089658353101550238585, −0.74874465690069088045070283340, 2.73196363021510964051814124013, 3.75593942941943279104065381582, 4.11221009080204364080623352365, 5.74067256078537240594826954001, 6.36077320795882412956182026918, 7.60166235638261270378635439947, 8.496157835515867998195351830788, 9.311441209664962524199543993685, 10.86036736195878182622979809676, 11.20174829155982719365439404653

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