Properties

Label 2-546-1.1-c5-0-56
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 − 54·5-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s − 216·10-s − 192·11-s + 144·12-s + 169·13-s + 196·14-s − 486·15-s + 256·16-s − 1.42e3·17-s + 324·18-s + 1.74e3·19-s − 864·20-s + 441·21-s − 768·22-s − 3.79e3·23-s + 576·24-s − 209·25-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 − 0.965·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.683·10-s − 0.478·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.557·15-s + 1/4·16-s − 1.19·17-s + 0.235·18-s + 1.11·19-s − 0.482·20-s + 0.218·21-s − 0.338·22-s − 1.49·23-s + 0.204·24-s − 0.0668·25-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 \( 1 + 54 T + p^{5} T^{2} \)
11 \( 1 + 192 T + p^{5} T^{2} \)
17 \( 1 + 1422 T + p^{5} T^{2} \)
19 \( 1 - 92 p T + p^{5} T^{2} \)
23 \( 1 + 3792 T + p^{5} T^{2} \)
29 \( 1 + 954 T + p^{5} T^{2} \)
31 \( 1 + 568 T + p^{5} T^{2} \)
37 \( 1 - 7886 T + p^{5} T^{2} \)
41 \( 1 + 14802 T + p^{5} T^{2} \)
43 \( 1 - 4964 T + p^{5} T^{2} \)
47 \( 1 + 18948 T + p^{5} T^{2} \)
53 \( 1 + 426 T + p^{5} T^{2} \)
59 \( 1 - 34872 T + p^{5} T^{2} \)
61 \( 1 + 25618 T + p^{5} T^{2} \)
67 \( 1 + 67060 T + p^{5} T^{2} \)
71 \( 1 + 28428 T + p^{5} T^{2} \)
73 \( 1 + 22894 T + p^{5} T^{2} \)
79 \( 1 + 1408 T + p^{5} T^{2} \)
83 \( 1 + 17304 T + p^{5} T^{2} \)
89 \( 1 + 93690 T + p^{5} T^{2} \)
97 \( 1 - 16826 T + p^{5} T^{2} \)
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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

−9.588799646227510532617139795388, −8.389873407751226519244235193178, −7.81727537196367218568248922072, −6.96133797171718847317187171148, −5.76391815936787233906114017714, −4.59495417357038157242404046543, −3.88047110823288346725278681275, −2.86004156254573473647139346450, −1.67238916731038755293690425631, 0, 1.67238916731038755293690425631, 2.86004156254573473647139346450, 3.88047110823288346725278681275, 4.59495417357038157242404046543, 5.76391815936787233906114017714, 6.96133797171718847317187171148, 7.81727537196367218568248922072, 8.389873407751226519244235193178, 9.588799646227510532617139795388

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