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Properties

Label 2-546-91.51-c1-0-2
Degree $2$
Conductor $546$
Sign $0.508 - 0.860i$
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  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−3.60 − 2.08i)5-s + 0.999i·6-s + (−2.59 − 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (2.08 + 3.60i)10-s + (1.87 − 1.08i)11-s + (0.499 − 0.866i)12-s + (−3.58 + 0.418i)13-s + (2 + 1.73i)14-s + 4.16i·15-s + (−0.5 + 0.866i)16-s + (0.581 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−1.61 − 0.930i)5-s + 0.408i·6-s + (−0.981 − 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.658 + 1.13i)10-s + (0.564 − 0.325i)11-s + (0.144 − 0.249i)12-s + (−0.993 + 0.116i)13-s + (0.534 + 0.462i)14-s + 1.07i·15-s + (−0.125 + 0.216i)16-s + (0.140 + 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.508 - 0.860i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.508 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151563 + 0.0864660i\)
\(L(\frac12)\) \(\approx\) \(0.151563 + 0.0864660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.59 + 0.5i)T \)
13 \( 1 + (3.58 - 0.418i)T \)
good5 \( 1 + (3.60 + 2.08i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.87 + 1.08i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.581 - 1.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.47 - 3.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.58 + 4.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.16T + 29T^{2} \)
31 \( 1 + (7.79 - 4.5i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.47 - 3.16i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2.83T + 43T^{2} \)
47 \( 1 + (4.75 + 2.74i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0811 + 0.140i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.06 - 4.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.74 + 3.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.93 + 4.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (5.47 - 3.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + (2.73 + 1.58i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11iT - 97T^{2} \)
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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

−11.20303061016276879915454704721, −9.981306916809689862735954878099, −9.122698735149658307976175817885, −8.279046780721271640015213306865, −7.45260013805559615572786716762, −6.80007979160693837239464314170, −5.31838061724140087491855776182, −4.06042517650664893329633287994, −3.13805737998588599360629300665, −1.10335649432289472779935953004, 0.15398078674611623472747333100, 2.91579499868153466910297387464, 3.77102553364251323183483507169, 5.05248253444049032372356695953, 6.34334831969580221988044801526, 7.35303622043527567807231232474, 7.56568572921026449195298488565, 9.164421386254993275624045107448, 9.614131456177503342707623885582, 10.63977139008519724902357857048

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