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Properties

Label 2-546-273.200-c1-0-17
Degree $2$
Conductor $546$
Sign $0.441 - 0.897i$
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 + (1.72 + 0.178i)3-s + (−0.866 − 0.499i)4-s + (0.250 − 0.935i)5-s + (−0.617 + 1.61i)6-s + (0.932 + 2.47i)7-s + (0.707 − 0.707i)8-s + (2.93 + 0.613i)9-s + (0.839 + 0.484i)10-s + (−0.265 − 0.989i)11-s + (−1.40 − 1.01i)12-s + (1.28 + 3.36i)13-s + (−2.63 + 0.259i)14-s + (0.598 − 1.56i)15-s + (0.500 + 0.866i)16-s + (2.05 − 3.56i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.994 + 0.102i)3-s + (−0.433 − 0.249i)4-s + (0.112 − 0.418i)5-s + (−0.252 + 0.660i)6-s + (0.352 + 0.935i)7-s + (0.249 − 0.249i)8-s + (0.978 + 0.204i)9-s + (0.265 + 0.153i)10-s + (−0.0799 − 0.298i)11-s + (−0.405 − 0.293i)12-s + (0.355 + 0.934i)13-s + (−0.703 + 0.0694i)14-s + (0.154 − 0.404i)15-s + (0.125 + 0.216i)16-s + (0.499 − 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62017 + 1.00835i\)
\(L(\frac12)\) \(\approx\) \(1.62017 + 1.00835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.178i)T \)
7 \( 1 + (-0.932 - 2.47i)T \)
13 \( 1 + (-1.28 - 3.36i)T \)
good5 \( 1 + (-0.250 + 0.935i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.265 + 0.989i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.05 + 3.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.67 + 0.715i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.318 + 0.551i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 + (-0.646 - 2.41i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.748 - 2.79i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.0484 + 0.0484i)T + 41iT^{2} \)
43 \( 1 + 8.67iT - 43T^{2} \)
47 \( 1 + (-3.23 - 0.867i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.87 + 3.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.50 + 5.60i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.594 - 1.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.436 + 1.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.55 + 8.55i)T + 71iT^{2} \)
73 \( 1 + (12.7 - 3.42i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0157 + 0.0272i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.06 + 6.06i)T + 83iT^{2} \)
89 \( 1 + (-6.23 - 1.67i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.64 - 8.64i)T - 97iT^{2} \)
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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

−10.78298449011874019216735735937, −9.663009355642037334749931417627, −8.844252606924465975355042274815, −8.606721070276512392274794992034, −7.49056095748524854231474503950, −6.55868458554894347918891999060, −5.32370230122045847086799928994, −4.48702237368646993506844267257, −3.08472159505722954615303404018, −1.67415503979320335326788120794, 1.30286429356237776625575977533, 2.63536736787404633736905626243, 3.67688177989293584511498745534, 4.52911585133784217329959626822, 6.16480299651749224778716526700, 7.43054992252788094085185649088, 8.016749068735821262261372986086, 8.862847035206334358311014422247, 10.12523340898781179481566203523, 10.32688742365806855955363175287

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