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Properties

Label 2-546-273.173-c1-0-11
Degree $2$
Conductor $546$
Sign $0.979 + 0.203i$
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  + (−0.5 − 0.866i)2-s + 1.73i·3-s + (−0.499 + 0.866i)4-s + (−3 − 1.73i)5-s + (1.49 − 0.866i)6-s + (−2.5 − 0.866i)7-s + 0.999·8-s − 2.99·9-s + 3.46i·10-s + 6·11-s + (−1.49 − 0.866i)12-s + (2.5 + 2.59i)13-s + (0.500 + 2.59i)14-s + (2.99 − 5.19i)15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 0.999i·3-s + (−0.249 + 0.433i)4-s + (−1.34 − 0.774i)5-s + (0.612 − 0.353i)6-s + (−0.944 − 0.327i)7-s + 0.353·8-s − 0.999·9-s + 1.09i·10-s + 1.80·11-s + (−0.433 − 0.249i)12-s + (0.693 + 0.720i)13-s + (0.133 + 0.694i)14-s + (0.774 − 1.34i)15-s + (−0.125 − 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869924 - 0.0894087i\)
\(L(\frac12)\) \(\approx\) \(0.869924 - 0.0894087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good5 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9 - 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 3.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.19iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92399237105760008284871008168, −9.637051559689543167790905495085, −9.206761792101193715469740189984, −8.526997109791615190189238041495, −7.35478119556352018881328624171, −6.18887108278425361161805108453, −4.58819167596123544806802341973, −3.92993539494884210594546864730, −3.30545436998788489064650558297, −0.898662168189236416027418993314, 0.916022881134693543895695025738, 3.04088424624477288477539624800, 3.89495838679659364257184706905, 5.78103876663881213874803509102, 6.50855481685002455101238226889, 7.23765733094344517964080420612, 7.890350884423884025705550180070, 8.901774658847514869591335329839, 9.667478339134525304217783277435, 11.21103068586605695850327550457

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