Properties

Label 2-546-1.1-c7-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 27·3-s + 64·4-s + 186.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s − 1.48e3·10-s − 1.33e3·11-s + 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 5.02e3·15-s + 4.09e3·16-s + 2.05e4·17-s − 5.83e3·18-s + 1.16e4·19-s + 1.19e4·20-s − 9.26e3·21-s + 1.06e4·22-s + 4.29e4·23-s − 1.38e4·24-s − 4.34e4·25-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 + 0.666·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.470·10-s − 0.302·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.384·15-s + 0.250·16-s + 1.01·17-s − 0.235·18-s + 0.390·19-s + 0.333·20-s − 0.218·21-s + 0.213·22-s + 0.736·23-s − 0.204·24-s − 0.556·25-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: \(0\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.358918951\)
\(L(\frac12)\) \(\approx\) \(2.358918951\)
\(L(\frac{9}{2})\) 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 \( 1 - 186.T + 7.81e4T^{2} \)
11 \( 1 + 1.33e3T + 1.94e7T^{2} \)
17 \( 1 - 2.05e4T + 4.10e8T^{2} \)
19 \( 1 - 1.16e4T + 8.93e8T^{2} \)
23 \( 1 - 4.29e4T + 3.40e9T^{2} \)
29 \( 1 + 2.10e5T + 1.72e10T^{2} \)
31 \( 1 - 2.47e5T + 2.75e10T^{2} \)
37 \( 1 + 1.84e5T + 9.49e10T^{2} \)
41 \( 1 - 3.48e5T + 1.94e11T^{2} \)
43 \( 1 + 2.39e5T + 2.71e11T^{2} \)
47 \( 1 + 3.64e5T + 5.06e11T^{2} \)
53 \( 1 - 1.66e6T + 1.17e12T^{2} \)
59 \( 1 + 2.42e6T + 2.48e12T^{2} \)
61 \( 1 - 1.85e6T + 3.14e12T^{2} \)
67 \( 1 - 4.86e6T + 6.06e12T^{2} \)
71 \( 1 + 9.58e5T + 9.09e12T^{2} \)
73 \( 1 + 2.97e6T + 1.10e13T^{2} \)
79 \( 1 + 3.17e5T + 1.92e13T^{2} \)
83 \( 1 - 4.87e6T + 2.71e13T^{2} \)
89 \( 1 - 1.72e6T + 4.42e13T^{2} \)
97 \( 1 - 1.57e6T + 8.07e13T^{2} \)
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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

−9.692213368501295558319806071163, −8.891579434502616864417878378873, −7.970466537218341566378529503841, −7.18994177767762460359144090577, −6.14121226468130288629094464152, −5.25197414379302353944380673993, −3.70436329117543321015842921995, −2.77333417597186984436927697719, −1.76116268641482602730359084769, −0.72896784250869987560480705609, 0.72896784250869987560480705609, 1.76116268641482602730359084769, 2.77333417597186984436927697719, 3.70436329117543321015842921995, 5.25197414379302353944380673993, 6.14121226468130288629094464152, 7.18994177767762460359144090577, 7.970466537218341566378529503841, 8.891579434502616864417878378873, 9.692213368501295558319806071163

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