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Properties

Label 2-546-91.5-c1-0-12
Degree $2$
Conductor $546$
Sign $0.292 + 0.956i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 + 0.499i)4-s + (−1.21 + 0.326i)5-s + (−0.707 − 0.707i)6-s + (−2.55 + 0.699i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.629 − 1.09i)10-s + (0.121 − 0.452i)11-s + (0.5 − 0.866i)12-s + (−0.447 − 3.57i)13-s + (−1.33 − 2.28i)14-s + (0.890 − 0.890i)15-s + (0.500 − 0.866i)16-s + (−1.64 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (−0.544 + 0.145i)5-s + (−0.288 − 0.288i)6-s + (−0.964 + 0.264i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.199 − 0.345i)10-s + (0.0365 − 0.136i)11-s + (0.144 − 0.249i)12-s + (−0.124 − 0.992i)13-s + (−0.356 − 0.610i)14-s + (0.230 − 0.230i)15-s + (0.125 − 0.216i)16-s + (−0.398 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285896 - 0.211420i\)
\(L(\frac12)\) \(\approx\) \(0.285896 - 0.211420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.55 - 0.699i)T \)
13 \( 1 + (0.447 + 3.57i)T \)
good5 \( 1 + (1.21 - 0.326i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.121 + 0.452i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.64 + 2.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.54 + 1.48i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.932 - 0.538i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 + (-2.51 + 9.37i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (8.93 - 2.39i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.51 + 5.51i)T + 41iT^{2} \)
43 \( 1 + 9.81iT - 43T^{2} \)
47 \( 1 + (-1.60 - 5.97i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.26 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.9 + 2.94i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.92 + 1.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.47 - 1.46i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.53 - 8.53i)T - 71iT^{2} \)
73 \( 1 + (5.23 + 1.40i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.70 + 8.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.71 + 7.71i)T + 83iT^{2} \)
89 \( 1 + (-1.58 - 5.93i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.33 + 6.33i)T + 97iT^{2} \)
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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

−10.56631855792295560707548731032, −9.630802578821986917182518835290, −8.939032893180033534208369682277, −7.63005856868916150437992165400, −7.06421629178490728015737254135, −5.87733542888811314533492099719, −5.27118468515361723132644819935, −3.93423752406348555231450001969, −3.01127843410044133338491213091, −0.20781343232204982917243653860, 1.59229175128410818029818407929, 3.24696458099727862808508334100, 4.17096036546090376396232984592, 5.28710713028865575309943242610, 6.45901207227518092575329714386, 7.22288724721637021360599616589, 8.431449171601726412149555474990, 9.453865201954304891930369641907, 10.19383080998351205371217671092, 11.10391947158364912952227000371

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