Properties

Label 2-546-1.1-c7-0-61
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 − 17.2·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s − 138.·10-s − 6.36e3·11-s − 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 466.·15-s + 4.09e3·16-s + 3.14e4·17-s + 5.83e3·18-s + 4.66e4·19-s − 1.10e3·20-s + 9.26e3·21-s − 5.09e4·22-s − 3.62e4·23-s − 1.38e4·24-s − 7.78e4·25-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 − 0.0617·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.0436·10-s − 1.44·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.0356·15-s + 0.250·16-s + 1.55·17-s + 0.235·18-s + 1.56·19-s − 0.0308·20-s + 0.218·21-s − 1.01·22-s − 0.622·23-s − 0.204·24-s − 0.996·25-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 \( 1 + 17.2T + 7.81e4T^{2} \)
11 \( 1 + 6.36e3T + 1.94e7T^{2} \)
17 \( 1 - 3.14e4T + 4.10e8T^{2} \)
19 \( 1 - 4.66e4T + 8.93e8T^{2} \)
23 \( 1 + 3.62e4T + 3.40e9T^{2} \)
29 \( 1 - 4.28e4T + 1.72e10T^{2} \)
31 \( 1 - 9.60e4T + 2.75e10T^{2} \)
37 \( 1 - 2.46e5T + 9.49e10T^{2} \)
41 \( 1 - 2.92e4T + 1.94e11T^{2} \)
43 \( 1 + 2.90e5T + 2.71e11T^{2} \)
47 \( 1 - 9.85e5T + 5.06e11T^{2} \)
53 \( 1 + 1.38e6T + 1.17e12T^{2} \)
59 \( 1 + 1.70e6T + 2.48e12T^{2} \)
61 \( 1 + 2.60e6T + 3.14e12T^{2} \)
67 \( 1 + 3.89e6T + 6.06e12T^{2} \)
71 \( 1 - 1.70e6T + 9.09e12T^{2} \)
73 \( 1 - 2.08e6T + 1.10e13T^{2} \)
79 \( 1 - 4.85e6T + 1.92e13T^{2} \)
83 \( 1 - 1.90e6T + 2.71e13T^{2} \)
89 \( 1 + 1.67e6T + 4.42e13T^{2} \)
97 \( 1 + 4.89e6T + 8.07e13T^{2} \)
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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

−9.596625027691185357068371125658, −7.81964302073972039669433325772, −7.56823902410991943099636937732, −6.16713958931692623891388157746, −5.52133482121313142576231420959, −4.75022356494391240843844781522, −3.47907845415490385947207515399, −2.63945196606061817086364932122, −1.21312968634746809402689581564, 0, 1.21312968634746809402689581564, 2.63945196606061817086364932122, 3.47907845415490385947207515399, 4.75022356494391240843844781522, 5.52133482121313142576231420959, 6.16713958931692623891388157746, 7.56823902410991943099636937732, 7.81964302073972039669433325772, 9.596625027691185357068371125658

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