Properties

Label 2-546-1.1-c5-0-57
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $87.5695$
Root an. cond. $9.35786$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s − 16·5-s + 36·6-s − 49·7-s + 64·8-s + 81·9-s − 64·10-s − 114·11-s + 144·12-s − 169·13-s − 196·14-s − 144·15-s + 256·16-s + 538·17-s + 324·18-s − 536·19-s − 256·20-s − 441·21-s − 456·22-s − 4.59e3·23-s + 576·24-s − 2.86e3·25-s − 676·26-s + 729·27-s − 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.286·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.202·10-s − 0.284·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.165·15-s + 1/4·16-s + 0.451·17-s + 0.235·18-s − 0.340·19-s − 0.143·20-s − 0.218·21-s − 0.200·22-s − 1.81·23-s + 0.204·24-s − 0.918·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(87.5695\)
Root analytic conductor: \(9.35786\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 - p^{2} T \)
7 \( 1 + p^{2} T \)
13 \( 1 + p^{2} T \)
good5 \( 1 + 16 T + p^{5} T^{2} \)
11 \( 1 + 114 T + p^{5} T^{2} \)
17 \( 1 - 538 T + p^{5} T^{2} \)
19 \( 1 + 536 T + p^{5} T^{2} \)
23 \( 1 + 4596 T + p^{5} T^{2} \)
29 \( 1 - 1594 T + p^{5} T^{2} \)
31 \( 1 - 9364 T + p^{5} T^{2} \)
37 \( 1 + 12002 T + p^{5} T^{2} \)
41 \( 1 - 4928 T + p^{5} T^{2} \)
43 \( 1 + 14284 T + p^{5} T^{2} \)
47 \( 1 + 22262 T + p^{5} T^{2} \)
53 \( 1 + 474 T + p^{5} T^{2} \)
59 \( 1 + 4182 T + p^{5} T^{2} \)
61 \( 1 + 21830 T + p^{5} T^{2} \)
67 \( 1 - 20780 T + p^{5} T^{2} \)
71 \( 1 - 18682 T + p^{5} T^{2} \)
73 \( 1 + 37866 T + p^{5} T^{2} \)
79 \( 1 + 27840 T + p^{5} T^{2} \)
83 \( 1 + 101914 T + p^{5} T^{2} \)
89 \( 1 + 77644 T + p^{5} T^{2} \)
97 \( 1 + 60050 T + p^{5} T^{2} \)
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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

−9.851754364247989144447430386776, −8.435819324377670253822432922997, −7.83822127719836991178545218090, −6.76836560156937313579698943830, −5.87745029768817222911068808661, −4.68533717589679297265900060254, −3.77054644278128225523809492652, −2.83814754167294671741018492050, −1.72601641530235749163359466802, 0, 1.72601641530235749163359466802, 2.83814754167294671741018492050, 3.77054644278128225523809492652, 4.68533717589679297265900060254, 5.87745029768817222911068808661, 6.76836560156937313579698943830, 7.83822127719836991178545218090, 8.435819324377670253822432922997, 9.851754364247989144447430386776

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