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Properties

Label 2-546-21.5-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.968 + 0.249i$
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 + (1.20 − 1.24i)3-s + (0.499 + 0.866i)4-s + (−1.33 + 2.30i)5-s + (−1.66 + 0.472i)6-s + (−2.41 − 1.08i)7-s − 0.999i·8-s + (−0.0871 − 2.99i)9-s + (2.30 − 1.33i)10-s + (−0.470 + 0.271i)11-s + (1.67 + 0.423i)12-s − i·13-s + (1.54 + 2.14i)14-s + (1.25 + 4.44i)15-s + (−0.5 + 0.866i)16-s + (−3.17 − 5.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.696 − 0.717i)3-s + (0.249 + 0.433i)4-s + (−0.596 + 1.03i)5-s + (−0.680 + 0.192i)6-s + (−0.912 − 0.408i)7-s − 0.353i·8-s + (−0.0290 − 0.999i)9-s + (0.730 − 0.421i)10-s + (−0.141 + 0.0818i)11-s + (0.484 + 0.122i)12-s − 0.277i·13-s + (0.414 + 0.573i)14-s + (0.325 + 1.14i)15-s + (−0.125 + 0.216i)16-s + (−0.769 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0656996 - 0.518994i\)
\(L(\frac12)\) \(\approx\) \(0.0656996 - 0.518994i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.24i)T \)
7 \( 1 + (2.41 + 1.08i)T \)
13 \( 1 + iT \)
good5 \( 1 + (1.33 - 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.470 - 0.271i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.17 + 5.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.41 + 3.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.03 + 1.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.75iT - 29T^{2} \)
31 \( 1 + (-5.60 + 3.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.73 - 8.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.50T + 41T^{2} \)
43 \( 1 + 9.79T + 43T^{2} \)
47 \( 1 + (3.16 - 5.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.39 + 3.69i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.14 + 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 3.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.29 - 3.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (1.57 - 0.911i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.20 + 5.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.54T + 83T^{2} \)
89 \( 1 + (-6.59 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.22iT - 97T^{2} \)
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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

−10.26869760182764172976597025637, −9.581144366504510606498221883572, −8.550498659620646816674496399829, −7.73084485018162111510577730955, −6.82657042309706886720248398660, −6.48761638674765314130314733538, −4.24566430590351901712399811642, −3.09008600918587066351876931955, −2.44702290803789214730431308207, −0.31654705340668568554214671203, 2.01299132307459589451979354226, 3.60586311987239317525799811298, 4.50591627462709858284807416464, 5.69672593366581068154375838590, 6.77932320801505416973235653968, 8.125298263220077627404586422324, 8.607592717378539657499395421157, 9.153223808557713726819899852583, 10.20279536928349499094646443421, 10.79574389418540072131658560973

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