Properties

Label 2-546-13.4-c1-0-8
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 − 2.73i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.36 + 2.36i)10-s + (4.5 + 2.59i)11-s − 0.999·12-s + (−0.866 − 3.5i)13-s + 0.999·14-s + (2.36 + 1.36i)15-s + (−0.5 + 0.866i)16-s + (−2.86 − 4.96i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 1.22i·5-s + (0.353 − 0.204i)6-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.431 + 0.748i)10-s + (1.35 + 0.783i)11-s − 0.288·12-s + (−0.240 − 0.970i)13-s + 0.267·14-s + (0.610 + 0.352i)15-s + (−0.125 + 0.216i)16-s + (−0.695 − 1.20i)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

\(L(1)\) \(\approx\) \(0.452761 - 0.593982i\)
\(L(\frac12)\) \(\approx\) \(0.452761 - 0.593982i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73iT - 5T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.59 - 3.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.267 - 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.23 + 9.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.73iT - 31T^{2} \)
37 \( 1 + (2.83 + 1.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.33 - 0.767i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.63 + 4.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.92iT - 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 + (-4.26 + 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.133 + 0.232i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.29 - 5.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.46iT - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (9.40 + 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.29 + 1.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.26127448710590795608230693503, −9.674877994452105573837401069904, −8.936466076214601355523869414706, −8.232217419737605284150690538308, −6.96424564403085186636217068398, −5.90577736808479048162270825976, −4.68965757326708442835757912045, −3.92301951337008298282573745801, −2.22309253777135583377504159053, −0.55780402277734569359906106642, 1.60261328394993151849986562835, 3.06389081997465632949987871796, 4.42392860858712601916531713415, 6.21264242957605226308921733799, 6.59729671185748879788068475144, 7.10273666132993184357042529551, 8.564095833256093044872084689538, 9.036304681467112522465570928457, 10.45586276183464764239771865515, 10.83551012177052173210948570960

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