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Properties

Label 2-546-273.62-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.563 - 0.826i$
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  + (−0.5 − 0.866i)2-s + (−0.334 + 1.69i)3-s + (−0.499 + 0.866i)4-s − 2.94i·5-s + (1.63 − 0.559i)6-s + (−1.08 + 2.41i)7-s + 0.999·8-s + (−2.77 − 1.13i)9-s + (−2.55 + 1.47i)10-s + (1.48 + 2.57i)11-s + (−1.30 − 1.13i)12-s + (−2.33 − 2.75i)13-s + (2.63 − 0.272i)14-s + (5.01 + 0.987i)15-s + (−0.5 − 0.866i)16-s + (−2.27 + 3.93i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.193 + 0.981i)3-s + (−0.249 + 0.433i)4-s − 1.31i·5-s + (0.669 − 0.228i)6-s + (−0.408 + 0.912i)7-s + 0.353·8-s + (−0.925 − 0.379i)9-s + (−0.807 + 0.466i)10-s + (0.448 + 0.776i)11-s + (−0.376 − 0.329i)12-s + (−0.646 − 0.762i)13-s + (0.703 − 0.0727i)14-s + (1.29 + 0.254i)15-s + (−0.125 − 0.216i)16-s + (−0.551 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.563 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198645 + 0.376032i\)
\(L(\frac12)\) \(\approx\) \(0.198645 + 0.376032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.334 - 1.69i)T \)
7 \( 1 + (1.08 - 2.41i)T \)
13 \( 1 + (2.33 + 2.75i)T \)
good5 \( 1 + 2.94iT - 5T^{2} \)
11 \( 1 + (-1.48 - 2.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.27 - 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.30 - 5.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.262 - 0.151i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 + (5.19 - 2.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.22 - 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.53 - 7.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.77iT - 47T^{2} \)
53 \( 1 + 1.95iT - 53T^{2} \)
59 \( 1 + (5.73 + 3.30i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.23 - 3.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.91 - 3.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.69 + 8.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.10T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 5.68iT - 83T^{2} \)
89 \( 1 + (-12.8 + 7.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.31 - 9.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96311949215252382080794800362, −10.06648357983735756254023169941, −9.428097410760641771777730477991, −8.727367688740605613751383835435, −8.073067006347341458147835709992, −6.30434227188383372120451842237, −5.24899863457829711428487021193, −4.49186222611998779076301008424, −3.39652080051146928540116207898, −1.85540091980568962744122993706, 0.26900167070823902477118685591, 2.24461423736964676554996138966, 3.55506440691410531869940535101, 5.13711750444779753440140627028, 6.47794290264844810812785261094, 6.98500924525098295302528704419, 7.25463445841583301089902348963, 8.640984139051103060855447358373, 9.467657244391645785954479532059, 10.79390031363407540611775415758

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