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Properties

Label 2-546-13.4-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.265 + 0.964i$
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 + 0.732i·5-s + (−0.866 + 0.499i)6-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.366 − 0.633i)10-s + (−3.23 − 1.86i)11-s + 0.999·12-s + (−0.866 − 3.5i)13-s − 0.999·14-s + (0.633 + 0.366i)15-s + (−0.5 + 0.866i)16-s + (0.133 + 0.232i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 0.327i·5-s + (−0.353 + 0.204i)6-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.115 − 0.200i)10-s + (−0.974 − 0.562i)11-s + 0.288·12-s + (−0.240 − 0.970i)13-s − 0.267·14-s + (0.163 + 0.0945i)15-s + (−0.125 + 0.216i)16-s + (0.0324 + 0.0562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635970 - 0.834337i\)
\(L(\frac12)\) \(\approx\) \(0.635970 - 0.834337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.866 + 3.5i)T \)
good5 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 + (3.23 + 1.86i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.133 - 0.232i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.86 + 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 + (-1.09 - 0.633i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.06 + 3.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.366 - 0.633i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.46iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-0.803 + 0.464i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.86 - 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.1 + 6.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.90 + 1.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.53iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 + (-5.59 - 3.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.633 + 0.366i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.59523312018909124984450473437, −9.733505119810891250899862986161, −8.651450823490251330936945466905, −7.892806333294131667241497870368, −7.28455907419813849896668152094, −6.09624138311318828423350091838, −4.90708555079700418080452177215, −3.25466188175055117454751485025, −2.47473604089590103073627587386, −0.73515905067729712723373549967, 1.69309782111050767955684654096, 3.14365998135259039462833424574, 4.75952421466538603659828579620, 5.29322995276508959541377287710, 6.74340070011453172178382661878, 7.62487018256716504318075570008, 8.489447630764671948278579891865, 9.265591809228363280108101585243, 10.00675681240024276702927683935, 10.82068384370190443321838013037

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