Properties

Label 2-546-1.1-c3-0-21
Degree $2$
Conductor $546$
Sign $1$
Analytic cond. $32.2150$
Root an. cond. $5.67582$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s + 9·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 18·10-s + 62·11-s + 12·12-s − 13·13-s − 14·14-s + 27·15-s + 16·16-s − 16·17-s + 18·18-s + 79·19-s + 36·20-s − 21·21-s + 124·22-s − 155·23-s + 24·24-s − 44·25-s − 26·26-s + 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.804·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.569·10-s + 1.69·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.464·15-s + 1/4·16-s − 0.228·17-s + 0.235·18-s + 0.953·19-s + 0.402·20-s − 0.218·21-s + 1.20·22-s − 1.40·23-s + 0.204·24-s − 0.351·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.719788467\)
\(L(\frac12)\) \(\approx\) \(4.719788467\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
7 \( 1 + p T \)
13 \( 1 + p T \)
good5 \( 1 - 9 T + p^{3} T^{2} \)
11 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 + 16 T + p^{3} T^{2} \)
19 \( 1 - 79 T + p^{3} T^{2} \)
23 \( 1 + 155 T + p^{3} T^{2} \)
29 \( 1 - 51 T + p^{3} T^{2} \)
31 \( 1 - 243 T + p^{3} T^{2} \)
37 \( 1 - 412 T + p^{3} T^{2} \)
41 \( 1 + 406 T + p^{3} T^{2} \)
43 \( 1 + 103 T + p^{3} T^{2} \)
47 \( 1 - 429 T + p^{3} T^{2} \)
53 \( 1 + 169 T + p^{3} T^{2} \)
59 \( 1 - 320 T + p^{3} T^{2} \)
61 \( 1 + 614 T + p^{3} T^{2} \)
67 \( 1 - 258 T + p^{3} T^{2} \)
71 \( 1 + 264 T + p^{3} T^{2} \)
73 \( 1 + 121 T + p^{3} T^{2} \)
79 \( 1 + 967 T + p^{3} T^{2} \)
83 \( 1 + 679 T + p^{3} T^{2} \)
89 \( 1 - 1059 T + p^{3} T^{2} \)
97 \( 1 + 21 T + p^{3} T^{2} \)
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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

−10.09676875081580879783302892932, −9.687301052017913627540667414404, −8.727059813739549369961785687929, −7.56853505192019611412712318255, −6.49939869250736506745519092691, −5.94682664835536978361966910708, −4.56063371766411552206021481573, −3.65174307837280419107596728716, −2.49508048285391660506775410016, −1.33752500240726781740567136295, 1.33752500240726781740567136295, 2.49508048285391660506775410016, 3.65174307837280419107596728716, 4.56063371766411552206021481573, 5.94682664835536978361966910708, 6.49939869250736506745519092691, 7.56853505192019611412712318255, 8.727059813739549369961785687929, 9.687301052017913627540667414404, 10.09676875081580879783302892932

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