Properties

Label 2-546-91.19-c1-0-6
Degree $2$
Conductor $546$
Sign $0.533 + 0.845i$
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 i·3-s + (0.866 + 0.499i)4-s + (−4.03 + 1.08i)5-s + (−0.258 + 0.965i)6-s + (−0.0127 + 2.64i)7-s + (−0.707 − 0.707i)8-s − 9-s + 4.18·10-s + (−0.447 − 0.447i)11-s + (0.499 − 0.866i)12-s + (1.96 − 3.02i)13-s + (0.697 − 2.55i)14-s + (1.08 + 4.03i)15-s + (0.500 + 0.866i)16-s + (2.95 − 5.12i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s − 0.577i·3-s + (0.433 + 0.249i)4-s + (−1.80 + 0.483i)5-s + (−0.105 + 0.394i)6-s + (−0.00480 + 0.999i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 1.32·10-s + (−0.135 − 0.135i)11-s + (0.144 − 0.249i)12-s + (0.544 − 0.838i)13-s + (0.186 − 0.682i)14-s + (0.279 + 1.04i)15-s + (0.125 + 0.216i)16-s + (0.717 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594776 - 0.328093i\)
\(L(\frac12)\) \(\approx\) \(0.594776 - 0.328093i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (0.0127 - 2.64i)T \)
13 \( 1 + (-1.96 + 3.02i)T \)
good5 \( 1 + (4.03 - 1.08i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.447 + 0.447i)T + 11iT^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.74 - 1.74i)T + 19iT^{2} \)
23 \( 1 + (-4.45 + 2.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.692 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.43 + 5.35i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.583 - 2.17i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.22 + 0.327i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.40 - 1.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.859 - 3.20i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.96 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.12 + 11.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 12.3iT - 61T^{2} \)
67 \( 1 + (-4.49 + 4.49i)T - 67iT^{2} \)
71 \( 1 + (-14.8 - 3.98i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.01 - 2.14i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.213 - 0.370i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.72 + 5.72i)T + 83iT^{2} \)
89 \( 1 + (1.62 + 0.434i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.95 + 7.30i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.03704741156359897598952922523, −9.710715523014801875063097742350, −8.616401920130814033319053715826, −7.975693346032835311833996020191, −7.40154379605008028490405825261, −6.37147307214255479368186906110, −5.08604231869122215644332498718, −3.44579269610234739957354145355, −2.74762318639711097981704331120, −0.65881265748877335669892677620, 1.03782954316947637984939754965, 3.47540341425080133347938020212, 4.10108950351655265633738383102, 5.18467420172221415719609340639, 6.78676919764501478041455569804, 7.50945824146726365903657166675, 8.332284692620252589587712354536, 8.979336226051489582203541432989, 10.13136102811677816404522241879, 10.93351921386728407482681878712

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