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Properties

Label 2-546-273.251-c1-0-12
Degree $2$
Conductor $546$
Sign $0.935 - 0.353i$
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 + (−1.19 − 1.25i)3-s + (−0.499 − 0.866i)4-s + 0.465i·5-s + (1.68 − 0.405i)6-s + (2.62 + 0.293i)7-s + 0.999·8-s + (−0.151 + 2.99i)9-s + (−0.403 − 0.232i)10-s + (−1.39 + 2.41i)11-s + (−0.490 + 1.66i)12-s + (−0.799 − 3.51i)13-s + (−1.56 + 2.13i)14-s + (0.584 − 0.555i)15-s + (−0.5 + 0.866i)16-s + (−0.508 − 0.880i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.689 − 0.724i)3-s + (−0.249 − 0.433i)4-s + 0.208i·5-s + (0.687 − 0.165i)6-s + (0.993 + 0.111i)7-s + 0.353·8-s + (−0.0504 + 0.998i)9-s + (−0.127 − 0.0736i)10-s + (−0.421 + 0.729i)11-s + (−0.141 + 0.479i)12-s + (−0.221 − 0.975i)13-s + (−0.419 + 0.569i)14-s + (0.150 − 0.143i)15-s + (−0.125 + 0.216i)16-s + (−0.123 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.935 - 0.353i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999690 + 0.182788i\)
\(L(\frac12)\) \(\approx\) \(0.999690 + 0.182788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.25i)T \)
7 \( 1 + (-2.62 - 0.293i)T \)
13 \( 1 + (0.799 + 3.51i)T \)
good5 \( 1 - 0.465iT - 5T^{2} \)
11 \( 1 + (1.39 - 2.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.508 + 0.880i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.817 + 1.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.14 - 4.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.01 - 2.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.94T + 31T^{2} \)
37 \( 1 + (-4.51 - 2.60i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.35 - 5.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.24 + 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.277iT - 47T^{2} \)
53 \( 1 + 8.99iT - 53T^{2} \)
59 \( 1 + (8.80 - 5.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.2 + 6.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.308 + 0.178i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.01 + 8.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.46T + 73T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 + (10.1 + 5.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.82 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.87888481075312968748102202504, −10.10638068175311806371348305761, −8.849151913189728943910973970522, −7.896138585792672928021281113756, −7.32874926414894621641278953239, −6.46205389791452383702068499433, −5.21845826285528047275486719774, −4.83764256952021326729914936360, −2.60907255620140842422411779249, −1.08832011172394646460161357213, 0.996770540931943795059665157623, 2.75332179174548966271372722582, 4.25724729280465955408182281578, 4.81417755358832906455267711312, 6.00309458454538089288605727460, 7.20329703273511734297274423936, 8.479958435414300928631543291887, 8.998791921967237659447305968025, 10.07460931531172674442011872785, 10.96284955568813922899359924036

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