Properties

Label 2-546-13.10-c1-0-1
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: $\chi_{546} (127, \cdot )$
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

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

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