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Properties

Label 2-546-273.251-c1-0-11
Degree $2$
Conductor $546$
Sign $0.889 - 0.457i$
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 + (−1.65 + 0.517i)3-s + (−0.499 − 0.866i)4-s − 0.188i·5-s + (0.377 − 1.69i)6-s + (−1.93 + 1.80i)7-s + 0.999·8-s + (2.46 − 1.71i)9-s + (0.163 + 0.0944i)10-s + (2.99 − 5.18i)11-s + (1.27 + 1.17i)12-s + (−3.59 − 0.247i)13-s + (−0.596 − 2.57i)14-s + (0.0978 + 0.312i)15-s + (−0.5 + 0.866i)16-s + (2.16 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.954 + 0.298i)3-s + (−0.249 − 0.433i)4-s − 0.0844i·5-s + (0.154 − 0.690i)6-s + (−0.731 + 0.682i)7-s + 0.353·8-s + (0.821 − 0.570i)9-s + (0.0517 + 0.0298i)10-s + (0.902 − 1.56i)11-s + (0.368 + 0.338i)12-s + (−0.997 − 0.0687i)13-s + (−0.159 − 0.688i)14-s + (0.0252 + 0.0806i)15-s + (−0.125 + 0.216i)16-s + (0.524 + 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.889 - 0.457i$
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: \((2,\ 546,\ (\ :1/2),\ 0.889 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749702 + 0.181460i\)
\(L(\frac12)\) \(\approx\) \(0.749702 + 0.181460i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 - 0.517i)T \)
7 \( 1 + (1.93 - 1.80i)T \)
13 \( 1 + (3.59 + 0.247i)T \)
good5 \( 1 + 0.188iT - 5T^{2} \)
11 \( 1 + (-2.99 + 5.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 + 4.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.72 - 3.30i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.73T + 31T^{2} \)
37 \( 1 + (-1.21 - 0.699i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.23 - 4.17i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.00 + 3.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.00iT - 47T^{2} \)
53 \( 1 - 0.440iT - 53T^{2} \)
59 \( 1 + (-1.61 + 0.929i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.69 - 0.976i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.40 - 5.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.71 + 2.96i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.64T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 16.1iT - 83T^{2} \)
89 \( 1 + (-12.3 - 7.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.15 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77619481189291804629650262061, −9.942265102750098813244105593732, −9.059256341289048452109522216782, −8.402775925424781413633716230818, −6.88371339385927861070385605562, −6.34251574852119626989845231945, −5.53535091114485558982101030538, −4.51873313796187292459474501311, −3.08689036735442203475659816814, −0.792628332008688945196826148046, 0.990141379074656472691957742751, 2.51444763071570069098831838679, 4.15804568725022812018682434947, 4.86267227404679042573819102111, 6.45034020589011563333568432447, 7.03400780490893994180777937855, 7.87645325561566901327755650252, 9.476234849999203643558300352866, 9.946130156280605166553692827314, 10.59745794423641276694876071419

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