Properties

Label 2-546-91.76-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.982 - 0.184i$
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 + (−2.46 + 2.46i)5-s + (−0.258 − 0.965i)6-s + (0.0731 − 2.64i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.74 + 3.01i)10-s + (−5.73 − 1.53i)11-s − 12-s + (−2.00 − 2.99i)13-s + (−2.53 − 0.755i)14-s + (−0.901 + 3.36i)15-s + (0.500 + 0.866i)16-s + (−0.681 + 1.18i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.10 + 1.10i)5-s + (−0.105 − 0.394i)6-s + (0.0276 − 0.999i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.550 + 0.953i)10-s + (−1.72 − 0.463i)11-s − 0.288·12-s + (−0.555 − 0.831i)13-s + (−0.677 − 0.201i)14-s + (−0.232 + 0.868i)15-s + (0.125 + 0.216i)16-s + (−0.165 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.982 - 0.184i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.982 - 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0518829 + 0.556838i\)
\(L(\frac12)\) \(\approx\) \(0.0518829 + 0.556838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0731 + 2.64i)T \)
13 \( 1 + (2.00 + 2.99i)T \)
good5 \( 1 + (2.46 - 2.46i)T - 5iT^{2} \)
11 \( 1 + (5.73 + 1.53i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.681 - 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.203 - 0.758i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.338 - 0.195i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.44 + 5.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.211 + 0.211i)T - 31iT^{2} \)
37 \( 1 + (5.84 + 1.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.49 + 0.936i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.10 - 1.21i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.84 - 6.84i)T + 47iT^{2} \)
53 \( 1 - 9.73T + 53T^{2} \)
59 \( 1 + (5.37 - 1.44i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.38 - 4.84i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.38 + 5.18i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-12.8 + 3.45i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.09 + 5.09i)T + 73iT^{2} \)
79 \( 1 - 9.25T + 79T^{2} \)
83 \( 1 + (9.33 - 9.33i)T - 83iT^{2} \)
89 \( 1 + (1.44 - 5.37i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.44 + 12.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.64391014380106544752645618493, −9.816574415919758008108638410814, −8.232173165364341330472723039716, −7.75130774333188183830441788586, −7.02486310454895394148631581194, −5.56294758572292214741468653404, −4.19123780313603353779130357315, −3.31434749221084710562348871771, −2.50130925370098829808170009899, −0.27011329600089133151610609825, 2.38984011363851161390729415722, 3.81334900088063262050718547240, 5.01454937140207291326004862048, 5.22424383915978810496427789238, 7.02737330638111234653115295802, 7.80131281701097044295362911044, 8.595051146824276894227108969902, 9.087465596516903795609974445754, 10.17570784085561498195526488432, 11.47489057411444272551101450264

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