Properties

Label 2-546-1.1-c7-0-16
Degree $2$
Conductor $546$
Sign $1$
Analytic cond. $170.562$
Root an. cond. $13.0599$
Motivic weight $7$
Arithmetic yes
Rational yes
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 − 390·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 3.12e3·10-s − 388·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 1.05e4·15-s + 4.09e3·16-s − 3.72e4·17-s + 5.83e3·18-s + 3.51e4·19-s − 2.49e4·20-s − 9.26e3·21-s − 3.10e3·22-s − 1.02e5·23-s + 1.38e4·24-s + 7.39e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.39·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.986·10-s − 0.0878·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.805·15-s + 1/4·16-s − 1.84·17-s + 0.235·18-s + 1.17·19-s − 0.697·20-s − 0.218·21-s − 0.0621·22-s − 1.76·23-s + 0.204·24-s + 0.946·25-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: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.464885150\)
\(L(\frac12)\) \(\approx\) \(2.464885150\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{3} T \)
3 \( 1 - p^{3} T \)
7 \( 1 + p^{3} T \)
13 \( 1 + p^{3} T \)
good5 \( 1 + 78 p T + p^{7} T^{2} \)
11 \( 1 + 388 T + p^{7} T^{2} \)
17 \( 1 + 37294 T + p^{7} T^{2} \)
19 \( 1 - 35164 T + p^{7} T^{2} \)
23 \( 1 + 102980 T + p^{7} T^{2} \)
29 \( 1 - 224826 T + p^{7} T^{2} \)
31 \( 1 + 150552 T + p^{7} T^{2} \)
37 \( 1 - 306058 T + p^{7} T^{2} \)
41 \( 1 - 784994 T + p^{7} T^{2} \)
43 \( 1 + 771532 T + p^{7} T^{2} \)
47 \( 1 - 653976 T + p^{7} T^{2} \)
53 \( 1 + 6646 T + p^{7} T^{2} \)
59 \( 1 - 1376600 T + p^{7} T^{2} \)
61 \( 1 + 1215494 T + p^{7} T^{2} \)
67 \( 1 + 3041808 T + p^{7} T^{2} \)
71 \( 1 - 611256 T + p^{7} T^{2} \)
73 \( 1 - 3531686 T + p^{7} T^{2} \)
79 \( 1 + 1351792 T + p^{7} T^{2} \)
83 \( 1 + 4882216 T + p^{7} T^{2} \)
89 \( 1 - 6893754 T + p^{7} T^{2} \)
97 \( 1 - 3768126 T + p^{7} T^{2} \)
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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

−9.652610957003786662847572155825, −8.597202559325568974251529179192, −7.77478886825009108727656558453, −7.08103789493710362863281259565, −6.07266470092761179161888304233, −4.63299561557641848505644228271, −4.06532565059314959409107099440, −3.13430665660150287474901555369, −2.17615530135503955002689924909, −0.56592606313912103428729292961, 0.56592606313912103428729292961, 2.17615530135503955002689924909, 3.13430665660150287474901555369, 4.06532565059314959409107099440, 4.63299561557641848505644228271, 6.07266470092761179161888304233, 7.08103789493710362863281259565, 7.77478886825009108727656558453, 8.597202559325568974251529179192, 9.652610957003786662847572155825

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