Properties

Label 2-546-1.1-c7-0-31
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 + 496.·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s + 3.97e3·10-s − 3.82e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 1.34e4·15-s + 4.09e3·16-s − 2.14e3·17-s + 5.83e3·18-s − 1.85e4·19-s + 3.18e4·20-s + 9.26e3·21-s − 3.05e4·22-s − 6.75e4·23-s − 1.38e4·24-s + 1.68e5·25-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 + 1.77·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.25·10-s − 0.866·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 1.02·15-s + 0.250·16-s − 0.106·17-s + 0.235·18-s − 0.620·19-s + 0.888·20-s + 0.218·21-s − 0.612·22-s − 1.15·23-s − 0.204·24-s + 2.16·25-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(4.152898077\)
\(L(\frac12)\) \(\approx\) \(4.152898077\)
\(L(\frac{9}{2})\) 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 \( 1 - 496.T + 7.81e4T^{2} \)
11 \( 1 + 3.82e3T + 1.94e7T^{2} \)
17 \( 1 + 2.14e3T + 4.10e8T^{2} \)
19 \( 1 + 1.85e4T + 8.93e8T^{2} \)
23 \( 1 + 6.75e4T + 3.40e9T^{2} \)
29 \( 1 - 5.87e4T + 1.72e10T^{2} \)
31 \( 1 - 1.57e5T + 2.75e10T^{2} \)
37 \( 1 - 3.81e5T + 9.49e10T^{2} \)
41 \( 1 - 2.27e5T + 1.94e11T^{2} \)
43 \( 1 - 5.32e5T + 2.71e11T^{2} \)
47 \( 1 - 7.14e5T + 5.06e11T^{2} \)
53 \( 1 + 3.52e5T + 1.17e12T^{2} \)
59 \( 1 + 5.34e5T + 2.48e12T^{2} \)
61 \( 1 - 9.47e5T + 3.14e12T^{2} \)
67 \( 1 - 2.20e6T + 6.06e12T^{2} \)
71 \( 1 + 5.03e6T + 9.09e12T^{2} \)
73 \( 1 - 1.46e6T + 1.10e13T^{2} \)
79 \( 1 - 8.61e6T + 1.92e13T^{2} \)
83 \( 1 + 8.80e6T + 2.71e13T^{2} \)
89 \( 1 - 3.11e6T + 4.42e13T^{2} \)
97 \( 1 - 2.00e6T + 8.07e13T^{2} \)
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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

−10.03172431627339295695890855243, −8.964596857522459995945018859223, −7.68017054314935014675435478106, −6.36690088782038673274210091440, −6.08146777249720056357861997276, −5.23994311264526848513935228969, −4.26252800408652070030667469964, −2.73635179864832842677227804766, −2.05228556972014238253990509680, −0.810002406834040888644411851669, 0.810002406834040888644411851669, 2.05228556972014238253990509680, 2.73635179864832842677227804766, 4.26252800408652070030667469964, 5.23994311264526848513935228969, 6.08146777249720056357861997276, 6.36690088782038673274210091440, 7.68017054314935014675435478106, 8.964596857522459995945018859223, 10.03172431627339295695890855243

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