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Properties

Label 2-546-13.10-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.125 + 0.992i$
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 + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − 4.39i·5-s + (−0.866 − 0.499i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (2.19 + 3.80i)10-s + (0.971 − 0.560i)11-s + 0.999·12-s + (−2.14 + 2.89i)13-s + 0.999·14-s + (3.80 − 2.19i)15-s + (−0.5 − 0.866i)16-s + (1.57 − 2.73i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 1.96i·5-s + (−0.353 − 0.204i)6-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.695 + 1.20i)10-s + (0.292 − 0.169i)11-s + 0.288·12-s + (−0.594 + 0.804i)13-s + 0.267·14-s + (0.983 − 0.567i)15-s + (−0.125 − 0.216i)16-s + (0.382 − 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.125 + 0.992i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.125 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531693 - 0.603202i\)
\(L(\frac12)\) \(\approx\) \(0.531693 - 0.603202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (2.14 - 2.89i)T \)
good5 \( 1 + 4.39iT - 5T^{2} \)
11 \( 1 + (-0.971 + 0.560i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.57 + 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.99 + 4.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.70 + 4.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.37 - 5.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.58iT - 31T^{2} \)
37 \( 1 + (-6.59 + 3.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.78 - 1.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.792 + 1.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.58iT - 47T^{2} \)
53 \( 1 - 8.44T + 53T^{2} \)
59 \( 1 + (6.30 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.56 + 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.51 - 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 3.52T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + (-9.02 + 5.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.06 - 0.613i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.19853274387614635027187108395, −9.418473840695402965022613715958, −8.832671635593677428700793693807, −8.289438532489119511147075135147, −7.08623207512350542404412384331, −5.90746684075281469389939863503, −4.77292817459108798808381240618, −4.20994325428367096955048700307, −2.18757437735285719857719446244, −0.52292639274108066341199473540, 2.02637429201744097395803530814, 2.93915415698978448934695211243, 3.84351538409597409447509703236, 6.07680318643072923847773376684, 6.53554268829025319815480270812, 7.66035682867447062966706234290, 8.091842788772472093892104082509, 9.539397403237703939939223798823, 10.26137862603720406908648546682, 10.77026099675574182421815935389

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