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Properties

Label 2-546-273.38-c1-0-19
Degree $2$
Conductor $546$
Sign $0.269 + 0.963i$
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 + (−1.71 − 0.270i)3-s + (−0.499 + 0.866i)4-s + (−2.20 + 1.27i)5-s + (−0.620 − 1.61i)6-s + (−2.32 + 1.25i)7-s − 0.999·8-s + (2.85 + 0.926i)9-s + (−2.20 − 1.27i)10-s + (2.05 − 3.55i)11-s + (1.08 − 1.34i)12-s + (−0.691 + 3.53i)13-s + (−2.25 − 1.38i)14-s + (4.11 − 1.57i)15-s + (−0.5 − 0.866i)16-s + (2.85 − 4.93i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.987 − 0.156i)3-s + (−0.249 + 0.433i)4-s + (−0.985 + 0.568i)5-s + (−0.253 − 0.660i)6-s + (−0.879 + 0.475i)7-s − 0.353·8-s + (0.951 + 0.308i)9-s + (−0.696 − 0.402i)10-s + (0.619 − 1.07i)11-s + (0.314 − 0.388i)12-s + (−0.191 + 0.981i)13-s + (−0.602 − 0.370i)14-s + (1.06 − 0.407i)15-s + (−0.125 − 0.216i)16-s + (0.691 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.269 + 0.963i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.269 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265703 - 0.201607i\)
\(L(\frac12)\) \(\approx\) \(0.265703 - 0.201607i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.270i)T \)
7 \( 1 + (2.32 - 1.25i)T \)
13 \( 1 + (0.691 - 3.53i)T \)
good5 \( 1 + (2.20 - 1.27i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.05 + 3.55i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.85 + 4.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.22 + 7.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 0.653i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.06iT - 29T^{2} \)
31 \( 1 + (-0.00131 + 0.00227i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.13 - 3.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.51iT - 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 + (3.68 - 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.39 + 2.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.68 + 0.971i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.80 - 1.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.68T + 71T^{2} \)
73 \( 1 + (-1.31 + 2.28i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.07 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + (3.31 - 1.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.91T + 97T^{2} \)
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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

−10.99638618922019637507920869363, −9.648035610106210606658748810991, −8.815714718654994440625950587143, −7.56383828464765565208786716526, −6.71794359403811117556079831692, −6.31819429994071304162672237340, −5.07528278458641884243011443609, −4.06043074480777410943350751940, −2.92105175606034187905939741476, −0.20822289646342447128854452408, 1.40144556222193464494846820013, 3.67417317666495159217662068576, 4.08526300237500614976725397673, 5.27081148501555906759535487967, 6.24441814767156780400584613336, 7.27458014004399488102401657540, 8.318567119592840641653251873106, 9.642825795016679852401168627319, 10.31563149240482060508949306457, 10.89526414437505969455996387214

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