Properties

Label 2-546-1.1-c7-0-30
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 + 287.·5-s − 216·6-s + 343·7-s + 512·8-s + 729·9-s + 2.29e3·10-s − 3.51e3·11-s − 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 7.75e3·15-s + 4.09e3·16-s + 1.09e4·17-s + 5.83e3·18-s − 2.58e4·19-s + 1.83e4·20-s − 9.26e3·21-s − 2.81e4·22-s + 6.40e4·23-s − 1.38e4·24-s + 4.47e3·25-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 + 1.02·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.727·10-s − 0.796·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.593·15-s + 0.250·16-s + 0.538·17-s + 0.235·18-s − 0.864·19-s + 0.514·20-s − 0.218·21-s − 0.563·22-s + 1.09·23-s − 0.204·24-s + 0.0572·25-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

\(L(4)\) \(\approx\) \(3.924441066\)
\(L(\frac12)\) \(\approx\) \(3.924441066\)
\(L(\frac{9}{2})\) 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 \( 1 - 287.T + 7.81e4T^{2} \)
11 \( 1 + 3.51e3T + 1.94e7T^{2} \)
17 \( 1 - 1.09e4T + 4.10e8T^{2} \)
19 \( 1 + 2.58e4T + 8.93e8T^{2} \)
23 \( 1 - 6.40e4T + 3.40e9T^{2} \)
29 \( 1 - 2.16e5T + 1.72e10T^{2} \)
31 \( 1 + 1.90e5T + 2.75e10T^{2} \)
37 \( 1 - 1.07e5T + 9.49e10T^{2} \)
41 \( 1 - 5.61e5T + 1.94e11T^{2} \)
43 \( 1 + 2.77e5T + 2.71e11T^{2} \)
47 \( 1 - 5.11e5T + 5.06e11T^{2} \)
53 \( 1 - 1.28e6T + 1.17e12T^{2} \)
59 \( 1 - 6.11e5T + 2.48e12T^{2} \)
61 \( 1 + 6.20e5T + 3.14e12T^{2} \)
67 \( 1 - 1.99e6T + 6.06e12T^{2} \)
71 \( 1 + 3.28e6T + 9.09e12T^{2} \)
73 \( 1 - 2.89e6T + 1.10e13T^{2} \)
79 \( 1 + 2.70e6T + 1.92e13T^{2} \)
83 \( 1 + 3.10e6T + 2.71e13T^{2} \)
89 \( 1 + 3.01e4T + 4.42e13T^{2} \)
97 \( 1 - 9.04e6T + 8.07e13T^{2} \)
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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

−9.977414411717689439461739672671, −8.825964875236507799433798261099, −7.65883353451274285917968466178, −6.72290694924626616246373567400, −5.78313662743102304425707704094, −5.20935833832867785202180927163, −4.29543132174734789620514758141, −2.84270768088735897565629248908, −1.96249410641756181039870745100, −0.796339897879458885890696262861, 0.796339897879458885890696262861, 1.96249410641756181039870745100, 2.84270768088735897565629248908, 4.29543132174734789620514758141, 5.20935833832867785202180927163, 5.78313662743102304425707704094, 6.72290694924626616246373567400, 7.65883353451274285917968466178, 8.825964875236507799433798261099, 9.977414411717689439461739672671

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