Properties

Label 2-546-1.1-c7-0-78
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 27·3-s + 64·4-s − 10·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 80·10-s + 1.50e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 270·15-s + 4.09e3·16-s − 1.04e3·17-s + 5.83e3·18-s − 9.06e3·19-s − 640·20-s − 9.26e3·21-s + 1.20e4·22-s − 9.89e4·23-s + 1.38e4·24-s − 7.80e4·25-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 − 0.0357·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.0252·10-s + 0.341·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0206·15-s + 1/4·16-s − 0.0514·17-s + 0.235·18-s − 0.303·19-s − 0.0178·20-s − 0.218·21-s + 0.241·22-s − 1.69·23-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: yes
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 \( 1 - p^{3} T \)
3 \( 1 - p^{3} T \)
7 \( 1 + p^{3} T \)
13 \( 1 - p^{3} T \)
good5 \( 1 + 2 p T + p^{7} T^{2} \)
11 \( 1 - 1508 T + p^{7} T^{2} \)
17 \( 1 + 1042 T + p^{7} T^{2} \)
19 \( 1 + 9068 T + p^{7} T^{2} \)
23 \( 1 + 98988 T + p^{7} T^{2} \)
29 \( 1 + 213642 T + p^{7} T^{2} \)
31 \( 1 + 22048 T + p^{7} T^{2} \)
37 \( 1 - 418246 T + p^{7} T^{2} \)
41 \( 1 + 76414 T + p^{7} T^{2} \)
43 \( 1 + 177524 T + p^{7} T^{2} \)
47 \( 1 - 631916 T + p^{7} T^{2} \)
53 \( 1 + 982354 T + p^{7} T^{2} \)
59 \( 1 + 596384 T + p^{7} T^{2} \)
61 \( 1 - 1863406 T + p^{7} T^{2} \)
67 \( 1 + 1845652 T + p^{7} T^{2} \)
71 \( 1 - 1632280 T + p^{7} T^{2} \)
73 \( 1 - 216650 T + p^{7} T^{2} \)
79 \( 1 + 6460168 T + p^{7} T^{2} \)
83 \( 1 - 3592152 T + p^{7} T^{2} \)
89 \( 1 + 9482662 T + p^{7} T^{2} \)
97 \( 1 - 840226 T + p^{7} T^{2} \)
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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

−9.323215567299887563768310128728, −8.194205921707959177733702930532, −7.43956525317654958554676863649, −6.34969379181831723846537577133, −5.62869530701833990314981776336, −4.20721176332782513868914523472, −3.69569971310356149828961154530, −2.49219681664845626921846534963, −1.57837734094700720670726401602, 0, 1.57837734094700720670726401602, 2.49219681664845626921846534963, 3.69569971310356149828961154530, 4.20721176332782513868914523472, 5.62869530701833990314981776336, 6.34969379181831723846537577133, 7.43956525317654958554676863649, 8.194205921707959177733702930532, 9.323215567299887563768310128728

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