Properties

Label 2-21e2-1.1-c3-0-6
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
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-s − 7·4-s − 12·5-s − 15·8-s − 12·10-s − 20·11-s − 84·13-s + 41·16-s + 96·17-s + 12·19-s + 84·20-s − 20·22-s + 176·23-s + 19·25-s − 84·26-s − 58·29-s − 264·31-s + 161·32-s + 96·34-s + 258·37-s + 12·38-s + 180·40-s + 156·43-s + 140·44-s + 176·46-s + 408·47-s + 19·50-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 1.07·5-s − 0.662·8-s − 0.379·10-s − 0.548·11-s − 1.79·13-s + 0.640·16-s + 1.36·17-s + 0.144·19-s + 0.939·20-s − 0.193·22-s + 1.59·23-s + 0.151·25-s − 0.633·26-s − 0.371·29-s − 1.52·31-s + 0.889·32-s + 0.484·34-s + 1.14·37-s + 0.0512·38-s + 0.711·40-s + 0.553·43-s + 0.479·44-s + 0.564·46-s + 1.26·47-s + 0.0537·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9555521974\)
\(L(\frac12)\) \(\approx\) \(0.9555521974\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + p^{3} T^{2} \)
5 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 + 20 T + p^{3} T^{2} \)
13 \( 1 + 84 T + p^{3} T^{2} \)
17 \( 1 - 96 T + p^{3} T^{2} \)
19 \( 1 - 12 T + p^{3} T^{2} \)
23 \( 1 - 176 T + p^{3} T^{2} \)
29 \( 1 + 2 p T + p^{3} T^{2} \)
31 \( 1 + 264 T + p^{3} T^{2} \)
37 \( 1 - 258 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 - 156 T + p^{3} T^{2} \)
47 \( 1 - 408 T + p^{3} T^{2} \)
53 \( 1 - 722 T + p^{3} T^{2} \)
59 \( 1 + 492 T + p^{3} T^{2} \)
61 \( 1 + 492 T + p^{3} T^{2} \)
67 \( 1 - 412 T + p^{3} T^{2} \)
71 \( 1 + 296 T + p^{3} T^{2} \)
73 \( 1 - 240 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 + 924 T + p^{3} T^{2} \)
89 \( 1 - 744 T + p^{3} T^{2} \)
97 \( 1 + 168 T + p^{3} T^{2} \)
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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

−10.72122548158270999567129565153, −9.708919889086927862769467591626, −8.965471597250687253587297922272, −7.69562535211020623542043492799, −7.39839396951072252540416293182, −5.58728155953764381900895438173, −4.89112699171750252770258870951, −3.86458773238389442839648134216, −2.82718636534777787812191212676, −0.57970681934057697198356574056, 0.57970681934057697198356574056, 2.82718636534777787812191212676, 3.86458773238389442839648134216, 4.89112699171750252770258870951, 5.58728155953764381900895438173, 7.39839396951072252540416293182, 7.69562535211020623542043492799, 8.965471597250687253587297922272, 9.708919889086927862769467591626, 10.72122548158270999567129565153

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