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Properties

Label 2-546-91.47-c1-0-10
Degree $2$
Conductor $546$
Sign $0.0372 + 0.999i$
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  + (−0.965 + 0.258i)2-s + (−0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (0.466 + 1.73i)5-s + (0.707 − 0.707i)6-s + (−0.0195 − 2.64i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.900 − 1.55i)10-s + (−3.06 − 0.821i)11-s + (−0.5 + 0.866i)12-s + (−2.35 − 2.72i)13-s + (0.703 + 2.55i)14-s + (−1.27 − 1.27i)15-s + (0.500 − 0.866i)16-s + (−0.111 − 0.193i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (0.208 + 0.777i)5-s + (0.288 − 0.288i)6-s + (−0.00738 − 0.999i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.284 − 0.493i)10-s + (−0.924 − 0.247i)11-s + (−0.144 + 0.249i)12-s + (−0.653 − 0.756i)13-s + (0.188 + 0.681i)14-s + (−0.328 − 0.328i)15-s + (0.125 − 0.216i)16-s + (−0.0270 − 0.0469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0372 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0372 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0372 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0372 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355025 - 0.342053i\)
\(L(\frac12)\) \(\approx\) \(0.355025 - 0.342053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.0195 + 2.64i)T \)
13 \( 1 + (2.35 + 2.72i)T \)
good5 \( 1 + (-0.466 - 1.73i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.06 + 0.821i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.111 + 0.193i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.205 - 0.767i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.65 + 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.94T + 29T^{2} \)
31 \( 1 + (7.00 + 1.87i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.56 + 5.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.12 + 7.12i)T - 41iT^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 + (0.218 - 0.0586i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.0417 - 0.0723i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.23 - 4.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.64 - 3.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.753 + 2.81i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (8.47 + 8.47i)T + 71iT^{2} \)
73 \( 1 + (1.28 - 4.80i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.88 + 5.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 + 2.38i)T - 83iT^{2} \)
89 \( 1 + (9.21 - 2.46i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.45 + 8.45i)T - 97iT^{2} \)
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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

−10.51497240441056643067173241046, −10.06846253826085111264145211710, −8.892759994793353348665109081888, −7.63456210957228267450591848806, −7.20921102870463398649626076789, −6.07334289193748538278176323128, −5.17369773248169275058261698335, −3.75032565120574836680255797404, −2.42140275509341357793724271108, −0.36833646672873547941559462282, 1.59292870884351173782918492631, 2.73961402075378234716027646490, 4.66432135130903486834290710796, 5.46924505319496425335366745718, 6.50586208027251403709340892998, 7.59605657806050750008377348086, 8.433864299226969777171670029838, 9.329241094575910569878152522394, 9.943384401890221376037250094258, 11.12491637994215717953536806203

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