Properties

Label 2-546-1.1-c5-0-24
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s + 81·5-s − 36·6-s − 49·7-s − 64·8-s + 81·9-s − 324·10-s + 191·11-s + 144·12-s − 169·13-s + 196·14-s + 729·15-s + 256·16-s − 871·17-s − 324·18-s − 479·19-s + 1.29e3·20-s − 441·21-s − 764·22-s + 1.38e3·23-s − 576·24-s + 3.43e3·25-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 + 1.44·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.02·10-s + 0.475·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.836·15-s + 1/4·16-s − 0.730·17-s − 0.235·18-s − 0.304·19-s + 0.724·20-s − 0.218·21-s − 0.336·22-s + 0.546·23-s − 0.204·24-s + 1.09·25-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: \(0\)
Selberg data: \((2,\ 546,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.628888283\)
\(L(\frac12)\) \(\approx\) \(2.628888283\)
\(L(\frac{7}{2})\) 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 \( 1 - 81 T + p^{5} T^{2} \)
11 \( 1 - 191 T + p^{5} T^{2} \)
17 \( 1 + 871 T + p^{5} T^{2} \)
19 \( 1 + 479 T + p^{5} T^{2} \)
23 \( 1 - 1387 T + p^{5} T^{2} \)
29 \( 1 + 5295 T + p^{5} T^{2} \)
31 \( 1 - 5940 T + p^{5} T^{2} \)
37 \( 1 - 13543 T + p^{5} T^{2} \)
41 \( 1 - 9464 T + p^{5} T^{2} \)
43 \( 1 - 17387 T + p^{5} T^{2} \)
47 \( 1 + 8112 T + p^{5} T^{2} \)
53 \( 1 - 18038 T + p^{5} T^{2} \)
59 \( 1 - 28784 T + p^{5} T^{2} \)
61 \( 1 - 14773 T + p^{5} T^{2} \)
67 \( 1 + 54354 T + p^{5} T^{2} \)
71 \( 1 - 64608 T + p^{5} T^{2} \)
73 \( 1 - 39461 T + p^{5} T^{2} \)
79 \( 1 + 95554 T + p^{5} T^{2} \)
83 \( 1 + 69634 T + p^{5} T^{2} \)
89 \( 1 + 51906 T + p^{5} T^{2} \)
97 \( 1 - 162654 T + p^{5} T^{2} \)
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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

−9.682043839969212839752049855829, −9.340579746985257965026867978795, −8.508820653170918059108345529054, −7.33572658058923665751512114192, −6.46389775620545961456139240223, −5.68949451960732118054349601943, −4.25543213601089819050079732899, −2.75828248314972333790553340551, −2.05675544531230108374108981300, −0.882814938169230378092230621802, 0.882814938169230378092230621802, 2.05675544531230108374108981300, 2.75828248314972333790553340551, 4.25543213601089819050079732899, 5.68949451960732118054349601943, 6.46389775620545961456139240223, 7.33572658058923665751512114192, 8.508820653170918059108345529054, 9.340579746985257965026867978795, 9.682043839969212839752049855829

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