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Properties

Label 2-546-273.272-c1-0-14
Degree $2$
Conductor $546$
Sign $0.403 + 0.914i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.403 + 0.914i$
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: \((2,\ 546,\ (\ :1/2),\ 0.403 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571181 - 0.372290i\)
\(L(\frac12)\) \(\approx\) \(0.571181 - 0.372290i\)
\(L(\frac{3}{2})\) 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

−10.45160032189966880332727225695, −10.22146276791407068110380045877, −8.881925507635537138551927321385, −7.77793903935510968103299455721, −6.96170404118510011159603907470, −6.43473293175940693314818377491, −5.07588362454133351970231125971, −4.02012577127890003000260425762, −2.20387788111732723482299890587, −0.65132659530273040693232371101, 1.23108914369029048240047038509, 2.91159062099861472238650177348, 4.60813431425811412251450161687, 5.54522309314487098559756689053, 6.24670922169680806689488592457, 7.51843077736097333524805247030, 8.311209703079798010391096609239, 9.425098465431428847436697587105, 9.976831652506918931828969093231, 10.94716874438464456474011225672

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