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Properties

Label 2-546-39.11-c1-0-14
Degree $2$
Conductor $546$
Sign $0.920 - 0.390i$
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 + (1.41 + 1.00i)3-s + (0.866 + 0.499i)4-s + (2.61 − 2.61i)5-s + (−1.10 − 1.33i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.989 + 2.83i)9-s + (−3.19 + 1.84i)10-s + (−1.56 + 5.82i)11-s + (0.721 + 1.57i)12-s + (1.58 + 3.23i)13-s − i·14-s + (6.30 − 1.06i)15-s + (0.500 + 0.866i)16-s + (1.62 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.815 + 0.578i)3-s + (0.433 + 0.249i)4-s + (1.16 − 1.16i)5-s + (−0.450 − 0.544i)6-s + (0.0978 + 0.365i)7-s + (−0.249 − 0.249i)8-s + (0.329 + 0.944i)9-s + (−1.01 + 0.583i)10-s + (−0.470 + 1.75i)11-s + (0.208 + 0.454i)12-s + (0.440 + 0.897i)13-s − 0.267i·14-s + (1.62 − 0.276i)15-s + (0.125 + 0.216i)16-s + (0.394 − 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65328 + 0.336325i\)
\(L(\frac12)\) \(\approx\) \(1.65328 + 0.336325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 1.00i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-1.58 - 3.23i)T \)
good5 \( 1 + (-2.61 + 2.61i)T - 5iT^{2} \)
11 \( 1 + (1.56 - 5.82i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.15 - 0.577i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.52 + 2.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.11 + 3.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.91 + 4.91i)T + 31iT^{2} \)
37 \( 1 + (-7.89 - 2.11i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.19 + 1.12i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.69 - 3.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.62 + 6.62i)T + 47iT^{2} \)
53 \( 1 + 4.59iT - 53T^{2} \)
59 \( 1 + (8.67 - 2.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.42 + 5.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.149 + 0.558i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.31 + 8.64i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.971 - 0.971i)T - 73iT^{2} \)
79 \( 1 + 4.53T + 79T^{2} \)
83 \( 1 + (1.86 - 1.86i)T - 83iT^{2} \)
89 \( 1 + (-1.68 + 6.30i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.38 - 0.908i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.35754664311131841524152107540, −9.697138369419162437345620250435, −9.318694491725572567948467379236, −8.480555918230942066139709592272, −7.63585296075845642560333948688, −6.33148796044530107995011722012, −5.02039577134737994315436117717, −4.32633467116014801953273610339, −2.43389680727283321206060400339, −1.76039237847261779369354845584, 1.29388340525396269778279218219, 2.71700064098420883827002081750, 3.39228455500391272742946836045, 5.77714319380097091039192977920, 6.22901790263179102651530805973, 7.25670765435803572916880347200, 8.145316521012740485524594230111, 8.820671792268975847904041897004, 9.904824121918769642538936059977, 10.62268076791942192890825683932

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