Properties

Label 2-546-273.272-c1-0-24
Degree $2$
Conductor $546$
Sign $0.787 - 0.616i$
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 + (1.68 + 0.420i)3-s + 4-s + 0.841i·5-s + (1.68 + 0.420i)6-s + (0.595 + 2.57i)7-s + 8-s + (2.64 + 1.41i)9-s + 0.841i·10-s + (1.68 + 0.420i)12-s + (−2.27 − 2.79i)13-s + (0.595 + 2.57i)14-s + (−0.354 + 1.41i)15-s + 16-s − 4.33·17-s + (2.64 + 1.41i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.970 + 0.242i)3-s + 0.5·4-s + 0.376i·5-s + (0.685 + 0.171i)6-s + (0.224 + 0.974i)7-s + 0.353·8-s + (0.881 + 0.471i)9-s + 0.266i·10-s + (0.485 + 0.121i)12-s + (−0.631 − 0.775i)13-s + (0.159 + 0.688i)14-s + (−0.0914 + 0.365i)15-s + 0.250·16-s − 1.05·17-s + (0.623 + 0.333i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.80302 + 0.966975i\)
\(L(\frac12)\) \(\approx\) \(2.80302 + 0.966975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.420i)T \)
7 \( 1 + (-0.595 - 2.57i)T \)
13 \( 1 + (2.27 + 2.79i)T \)
good5 \( 1 - 0.841iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 4.33T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 + 7.98iT - 23T^{2} \)
29 \( 1 - 2.32iT - 29T^{2} \)
31 \( 1 - 5.53T + 31T^{2} \)
37 \( 1 - 5.15iT - 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 7.82iT - 47T^{2} \)
53 \( 1 + 7.98iT - 53T^{2} \)
59 \( 1 - 3.91iT - 59T^{2} \)
61 \( 1 - 5.59iT - 61T^{2} \)
67 \( 1 - 1.82iT - 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 7.27iT - 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 - 9.87T + 97T^{2} \)
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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

−10.71015480852045844242544900767, −10.24955742744538919309268896709, −8.836331075844567002670077511706, −8.451090614270476053905031954730, −7.21082685768673495183527177902, −6.34701823415941782331989098951, −5.05934704975560259293049053304, −4.24145408803672970481761681498, −2.82438440641602750785963306709, −2.29183839840830192334422019742, 1.54778598981722576761583854287, 2.80658277856213348852764073029, 4.18957740480635236303461343863, 4.58696474229942634945946628329, 6.26901451934859886760030043825, 7.15178679724268468817933696333, 7.85898161962547923712390375094, 8.932247006779094078978649584837, 9.740743676129188863550700486403, 10.80431383629467638382035708358

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