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Properties

Label 2-546-273.257-c1-0-21
Degree $2$
Conductor $546$
Sign $0.682 - 0.730i$
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 + (0.942 + 1.45i)3-s + 4-s + (1.98 + 1.14i)5-s + (0.942 + 1.45i)6-s + (−0.877 − 2.49i)7-s + 8-s + (−1.22 + 2.73i)9-s + (1.98 + 1.14i)10-s + (0.148 − 0.257i)11-s + (0.942 + 1.45i)12-s + (3.20 + 1.66i)13-s + (−0.877 − 2.49i)14-s + (0.206 + 3.95i)15-s + 16-s − 0.893·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.544 + 0.838i)3-s + 0.5·4-s + (0.886 + 0.511i)5-s + (0.384 + 0.593i)6-s + (−0.331 − 0.943i)7-s + 0.353·8-s + (−0.407 + 0.913i)9-s + (0.626 + 0.361i)10-s + (0.0447 − 0.0775i)11-s + (0.272 + 0.419i)12-s + (0.887 + 0.460i)13-s + (−0.234 − 0.667i)14-s + (0.0532 + 1.02i)15-s + 0.250·16-s − 0.216·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.682 - 0.730i$
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: \((2,\ 546,\ (\ :1/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62244 + 1.13846i\)
\(L(\frac12)\) \(\approx\) \(2.62244 + 1.13846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.942 - 1.45i)T \)
7 \( 1 + (0.877 + 2.49i)T \)
13 \( 1 + (-3.20 - 1.66i)T \)
good5 \( 1 + (-1.98 - 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.148 + 0.257i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.893T + 17T^{2} \)
19 \( 1 + (3.94 + 6.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.81iT - 23T^{2} \)
29 \( 1 + (-0.980 + 0.566i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.839 + 1.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.99iT - 37T^{2} \)
41 \( 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.94 - 3.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.21 + 3.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.28 + 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.02iT - 59T^{2} \)
61 \( 1 + (-7.31 + 4.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.94 - 1.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.14 + 1.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.16 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.46 + 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.54iT - 83T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (1.19 - 2.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.97428233542505611000639964701, −10.04961018280694487605248279093, −9.410504709904262933075771830524, −8.327432258299674004645475164995, −7.05452343425522105870459774124, −6.31925842049856211235145222192, −5.17401968539215865825084712290, −4.11005698334916127612553460830, −3.28474805764479307098749777282, −2.05583974861488194597029492682, 1.60678819374170612393569077214, 2.57478309576721324273658018435, 3.79100256342636970401562184850, 5.32952521789441069623728707675, 6.11496560963523714088218728259, 6.69198482856114638044580476174, 8.269990808380860127782879789872, 8.626574122384846765782840884975, 9.749922723568169283277634584633, 10.72067939709848524693055189058

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