Properties

Label 2-546-1.1-c7-0-63
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $170.562$
Root an. cond. $13.0599$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 27·3-s + 64·4-s + 58.2·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s − 465.·10-s − 124.·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 1.57e3·15-s + 4.09e3·16-s + 1.13e4·17-s − 5.83e3·18-s − 2.24e4·19-s + 3.72e3·20-s − 9.26e3·21-s + 992.·22-s + 3.41e4·23-s − 1.38e4·24-s − 7.47e4·25-s + 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.208·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.147·10-s − 0.0280·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.120·15-s + 0.250·16-s + 0.560·17-s − 0.235·18-s − 0.749·19-s + 0.104·20-s − 0.218·21-s + 0.0198·22-s + 0.586·23-s − 0.204·24-s − 0.956·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(170.562\)
Root analytic conductor: \(13.0599\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8T \)
3 \( 1 - 27T \)
7 \( 1 + 343T \)
13 \( 1 + 2.19e3T \)
good5 \( 1 - 58.2T + 7.81e4T^{2} \)
11 \( 1 + 124.T + 1.94e7T^{2} \)
17 \( 1 - 1.13e4T + 4.10e8T^{2} \)
19 \( 1 + 2.24e4T + 8.93e8T^{2} \)
23 \( 1 - 3.41e4T + 3.40e9T^{2} \)
29 \( 1 - 6.26e4T + 1.72e10T^{2} \)
31 \( 1 - 2.94e4T + 2.75e10T^{2} \)
37 \( 1 - 4.78e5T + 9.49e10T^{2} \)
41 \( 1 + 3.58e5T + 1.94e11T^{2} \)
43 \( 1 + 6.32e5T + 2.71e11T^{2} \)
47 \( 1 - 1.27e6T + 5.06e11T^{2} \)
53 \( 1 + 4.96e5T + 1.17e12T^{2} \)
59 \( 1 + 7.18e5T + 2.48e12T^{2} \)
61 \( 1 + 2.04e6T + 3.14e12T^{2} \)
67 \( 1 + 1.23e6T + 6.06e12T^{2} \)
71 \( 1 + 2.76e5T + 9.09e12T^{2} \)
73 \( 1 - 1.97e6T + 1.10e13T^{2} \)
79 \( 1 - 3.05e6T + 1.92e13T^{2} \)
83 \( 1 + 9.16e6T + 2.71e13T^{2} \)
89 \( 1 + 1.04e7T + 4.42e13T^{2} \)
97 \( 1 + 7.32e6T + 8.07e13T^{2} \)
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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

−9.275781869947455458114876363945, −8.388998328664835955018036049486, −7.61777867170875123410382013137, −6.69137960608281422765040806715, −5.76001149590988408480127595055, −4.41768590798856777127727033138, −3.21556987609132093519166141349, −2.31973277515188267158548523423, −1.22767316588398858241725799898, 0, 1.22767316588398858241725799898, 2.31973277515188267158548523423, 3.21556987609132093519166141349, 4.41768590798856777127727033138, 5.76001149590988408480127595055, 6.69137960608281422765040806715, 7.61777867170875123410382013137, 8.388998328664835955018036049486, 9.275781869947455458114876363945

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