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Properties

Label 2-546-273.38-c1-0-18
Degree $2$
Conductor $546$
Sign $0.612 + 0.790i$
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 + (−1.71 − 0.270i)3-s + (−0.499 + 0.866i)4-s + (2.20 − 1.27i)5-s + (0.620 + 1.61i)6-s + (2.32 − 1.25i)7-s + 0.999·8-s + (2.85 + 0.926i)9-s + (−2.20 − 1.27i)10-s + (−2.05 + 3.55i)11-s + (1.08 − 1.34i)12-s + (0.691 + 3.53i)13-s + (−2.25 − 1.38i)14-s + (−4.11 + 1.57i)15-s + (−0.5 − 0.866i)16-s + (2.85 − 4.93i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.987 − 0.156i)3-s + (−0.249 + 0.433i)4-s + (0.985 − 0.568i)5-s + (0.253 + 0.660i)6-s + (0.879 − 0.475i)7-s + 0.353·8-s + (0.951 + 0.308i)9-s + (−0.696 − 0.402i)10-s + (−0.619 + 1.07i)11-s + (0.314 − 0.388i)12-s + (0.191 + 0.981i)13-s + (−0.602 − 0.370i)14-s + (−1.06 + 0.407i)15-s + (−0.125 − 0.216i)16-s + (0.691 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03525 - 0.507871i\)
\(L(\frac12)\) \(\approx\) \(1.03525 - 0.507871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.270i)T \)
7 \( 1 + (-2.32 + 1.25i)T \)
13 \( 1 + (-0.691 - 3.53i)T \)
good5 \( 1 + (-2.20 + 1.27i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.05 - 3.55i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.85 + 4.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.22 - 7.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 0.653i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.06iT - 29T^{2} \)
31 \( 1 + (0.00131 - 0.00227i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.13 + 3.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 + (-3.68 + 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.39 + 2.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 0.971i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.80 + 1.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (1.31 - 2.28i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.07 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (-3.31 + 1.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.91T + 97T^{2} \)
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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

−10.58777008303381503945759578390, −9.877434918251600269425716387622, −9.350562823342864374107335440566, −7.83953957020786168991656543584, −7.27441883261097902099195395289, −5.83454316370694098974011444111, −5.04446920521844927731142748780, −4.18839511190473417991692304230, −2.09072528469467256630240525220, −1.18834426909422279207360341913, 1.17198566391292515146216880798, 2.95899313987378559117255405115, 4.84028228061626437390178714669, 5.67752573186118397567066645847, 6.03997821871073886352793564066, 7.25996238599697286465065968185, 8.190436780286793512785527895904, 9.202736338426039320795303263664, 10.20827869353583989171123831929, 10.82340334200373642929342021308

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