Properties

Label 2-546-273.257-c1-0-8
Degree $2$
Conductor $546$
Sign $0.881 - 0.472i$
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  + 2-s + (−1.23 − 1.20i)3-s + 4-s + (1.58 + 0.917i)5-s + (−1.23 − 1.20i)6-s + (−0.364 + 2.62i)7-s + 8-s + (0.0726 + 2.99i)9-s + (1.58 + 0.917i)10-s + (−1.69 + 2.93i)11-s + (−1.23 − 1.20i)12-s + (3.07 + 1.88i)13-s + (−0.364 + 2.62i)14-s + (−0.860 − 3.06i)15-s + 16-s + 3.18·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.715 − 0.698i)3-s + 0.5·4-s + (0.711 + 0.410i)5-s + (−0.506 − 0.493i)6-s + (−0.137 + 0.990i)7-s + 0.353·8-s + (0.0242 + 0.999i)9-s + (0.502 + 0.290i)10-s + (−0.511 + 0.885i)11-s + (−0.357 − 0.349i)12-s + (0.851 + 0.523i)13-s + (−0.0973 + 0.700i)14-s + (−0.222 − 0.790i)15-s + 0.250·16-s + 0.772·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87727 + 0.471406i\)
\(L(\frac12)\) \(\approx\) \(1.87727 + 0.471406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.23 + 1.20i)T \)
7 \( 1 + (0.364 - 2.62i)T \)
13 \( 1 + (-3.07 - 1.88i)T \)
good5 \( 1 + (-1.58 - 0.917i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.69 - 2.93i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 + (1.01 + 1.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.47iT - 23T^{2} \)
29 \( 1 + (-1.81 + 1.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.14 + 5.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.503iT - 37T^{2} \)
41 \( 1 + (-4.35 + 2.51i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.528 + 0.916i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.0 - 5.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.81 - 3.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.20iT - 59T^{2} \)
61 \( 1 + (-10.1 + 5.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.77 - 13.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.47 + 7.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.92 + 6.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 - 5.33iT - 89T^{2} \)
97 \( 1 + (4.79 - 8.31i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.06703336597315263885061586412, −10.21954835917216922171533273356, −9.214116488055646348716499635068, −7.900337502282098383030977543390, −6.99979765559139702100819704150, −5.99255013076113626067627787823, −5.65788702743534013177104335716, −4.42803835156488038272141758808, −2.70865913177282905361803430887, −1.81073361995733751002856846381, 1.07274280787145338552057579345, 3.21355683371857865981591697350, 4.06962102796135974849438672708, 5.27942707378643997727251462844, 5.80553192685957890780195238499, 6.74189374154631788794358325632, 8.023978644459282491534195483806, 9.135227467894187012731545203388, 10.31561963395312592376792526941, 10.58149570572964622092932350592

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