Properties

Label 2-546-1.1-c7-0-68
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 − 12.1·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 97.2·10-s + 2.35e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 328.·15-s + 4.09e3·16-s − 2.37e4·17-s − 5.83e3·18-s + 4.09e4·19-s − 777.·20-s + 9.26e3·21-s − 1.88e4·22-s − 3.67e4·23-s − 1.38e4·24-s − 7.79e4·25-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 − 0.0434·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.0307·10-s + 0.532·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0250·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 1.36·19-s − 0.0217·20-s + 0.218·21-s − 0.376·22-s − 0.629·23-s − 0.204·24-s − 0.998·25-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 \( 1 + 12.1T + 7.81e4T^{2} \)
11 \( 1 - 2.35e3T + 1.94e7T^{2} \)
17 \( 1 + 2.37e4T + 4.10e8T^{2} \)
19 \( 1 - 4.09e4T + 8.93e8T^{2} \)
23 \( 1 + 3.67e4T + 3.40e9T^{2} \)
29 \( 1 + 1.45e5T + 1.72e10T^{2} \)
31 \( 1 + 9.10e4T + 2.75e10T^{2} \)
37 \( 1 - 3.89e5T + 9.49e10T^{2} \)
41 \( 1 - 5.92e5T + 1.94e11T^{2} \)
43 \( 1 + 4.08e5T + 2.71e11T^{2} \)
47 \( 1 + 1.19e6T + 5.06e11T^{2} \)
53 \( 1 + 1.59e6T + 1.17e12T^{2} \)
59 \( 1 - 6.97e5T + 2.48e12T^{2} \)
61 \( 1 + 2.63e6T + 3.14e12T^{2} \)
67 \( 1 - 2.94e6T + 6.06e12T^{2} \)
71 \( 1 - 2.79e6T + 9.09e12T^{2} \)
73 \( 1 - 3.60e6T + 1.10e13T^{2} \)
79 \( 1 + 5.23e6T + 1.92e13T^{2} \)
83 \( 1 + 3.05e6T + 2.71e13T^{2} \)
89 \( 1 + 2.20e6T + 4.42e13T^{2} \)
97 \( 1 + 1.71e7T + 8.07e13T^{2} \)
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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

−9.436418970929010778693519673568, −8.283131754249614096352439242225, −7.69949947383785125403961690196, −6.73596319756174655550920599408, −5.70790499545174414802055932610, −4.36611303493675314780663792460, −3.36878125568649746244023792179, −2.14776007743985102504299177008, −1.32114274971617126046260370601, 0, 1.32114274971617126046260370601, 2.14776007743985102504299177008, 3.36878125568649746244023792179, 4.36611303493675314780663792460, 5.70790499545174414802055932610, 6.73596319756174655550920599408, 7.69949947383785125403961690196, 8.283131754249614096352439242225, 9.436418970929010778693519673568

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