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Properties

Label 2-546-273.257-c1-0-18
Degree $2$
Conductor $546$
Sign $0.536 - 0.843i$
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 + (0.517 + 1.65i)3-s + 4-s + (0.870 + 0.502i)5-s + (0.517 + 1.65i)6-s + (2.64 + 0.0151i)7-s + 8-s + (−2.46 + 1.71i)9-s + (0.870 + 0.502i)10-s + (−0.310 + 0.537i)11-s + (0.517 + 1.65i)12-s + (−1.14 − 3.41i)13-s + (2.64 + 0.0151i)14-s + (−0.380 + 1.69i)15-s + 16-s + 0.342·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.298 + 0.954i)3-s + 0.5·4-s + (0.389 + 0.224i)5-s + (0.211 + 0.674i)6-s + (0.999 + 0.00572i)7-s + 0.353·8-s + (−0.821 + 0.570i)9-s + (0.275 + 0.158i)10-s + (−0.0935 + 0.162i)11-s + (0.149 + 0.477i)12-s + (−0.316 − 0.948i)13-s + (0.707 + 0.00404i)14-s + (−0.0981 + 0.438i)15-s + 0.250·16-s + 0.0831·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.536 - 0.843i$
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: \((2,\ 546,\ (\ :1/2),\ 0.536 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35195 + 1.29103i\)
\(L(\frac12)\) \(\approx\) \(2.35195 + 1.29103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.517 - 1.65i)T \)
7 \( 1 + (-2.64 - 0.0151i)T \)
13 \( 1 + (1.14 + 3.41i)T \)
good5 \( 1 + (-0.870 - 0.502i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.310 - 0.537i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.342T + 17T^{2} \)
19 \( 1 + (-2.16 - 3.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (8.23 - 4.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.34iT - 37T^{2} \)
41 \( 1 + (-7.47 + 4.31i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.602 - 1.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0442 - 0.0255i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.15 - 2.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.08iT - 59T^{2} \)
61 \( 1 + (-5.78 + 3.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.75 + 2.74i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.621 - 1.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.46 + 7.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.458 + 0.793i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 - 3.60iT - 89T^{2} \)
97 \( 1 + (5.10 - 8.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82063022896261079524441101804, −10.28494419985807176782211431659, −9.329914740492921559402461754831, −8.167548384169924769789819926132, −7.50283027091694526037221438991, −5.92891917288226474973162921607, −5.26733535979509451328681098773, −4.34443936182808122445708953294, −3.25352604213497076522700545731, −2.07023141257072071309904765265, 1.49736473607590016577676537514, 2.46715198442505739132399954789, 3.93109452029502828074714517347, 5.18281402828185748282385122251, 5.93689368659463580725152849732, 7.16520750625400543704706384628, 7.66828185366850206439718088880, 8.837572296784595565919667477396, 9.610424436527337458007734127002, 11.33436574877215479409896010748

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