Properties

Label 2-21e2-1.1-c3-0-20
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  − 4·2-s + 8·4-s + 18·5-s − 72·10-s + 50·11-s + 36·13-s − 64·16-s + 126·17-s + 72·19-s + 144·20-s − 200·22-s − 14·23-s + 199·25-s − 144·26-s − 158·29-s + 36·31-s + 256·32-s − 504·34-s − 162·37-s − 288·38-s − 270·41-s − 324·43-s + 400·44-s + 56·46-s − 72·47-s − 796·50-s + 288·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.60·5-s − 2.27·10-s + 1.37·11-s + 0.768·13-s − 16-s + 1.79·17-s + 0.869·19-s + 1.60·20-s − 1.93·22-s − 0.126·23-s + 1.59·25-s − 1.08·26-s − 1.01·29-s + 0.208·31-s + 1.41·32-s − 2.54·34-s − 0.719·37-s − 1.22·38-s − 1.02·41-s − 1.14·43-s + 1.37·44-s + 0.179·46-s − 0.223·47-s − 2.25·50-s + 0.768·52-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

\(L(2)\) \(\approx\) \(1.546389654\)
\(L(\frac12)\) \(\approx\) \(1.546389654\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{2} T + p^{3} T^{2} \)
5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 50 T + p^{3} T^{2} \)
13 \( 1 - 36 T + p^{3} T^{2} \)
17 \( 1 - 126 T + p^{3} T^{2} \)
19 \( 1 - 72 T + p^{3} T^{2} \)
23 \( 1 + 14 T + p^{3} T^{2} \)
29 \( 1 + 158 T + p^{3} T^{2} \)
31 \( 1 - 36 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 + 270 T + p^{3} T^{2} \)
43 \( 1 + 324 T + p^{3} T^{2} \)
47 \( 1 + 72 T + p^{3} T^{2} \)
53 \( 1 - 22 T + p^{3} T^{2} \)
59 \( 1 - 468 T + p^{3} T^{2} \)
61 \( 1 + 792 T + p^{3} T^{2} \)
67 \( 1 - 232 T + p^{3} T^{2} \)
71 \( 1 - 734 T + p^{3} T^{2} \)
73 \( 1 + 180 T + p^{3} T^{2} \)
79 \( 1 - 236 T + p^{3} T^{2} \)
83 \( 1 - 36 T + p^{3} T^{2} \)
89 \( 1 - 234 T + p^{3} T^{2} \)
97 \( 1 + 468 T + p^{3} T^{2} \)
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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

−10.23033031976023580655049014979, −9.721459430642370501499305595259, −9.138265308090915139649259479095, −8.249119455653192645744076477598, −7.09547112697851912332151090138, −6.20830695344423707404086769149, −5.26215706401231760157048232762, −3.45862768474983275245374541917, −1.75354957020147024898294374465, −1.13271064717039472639225858843, 1.13271064717039472639225858843, 1.75354957020147024898294374465, 3.45862768474983275245374541917, 5.26215706401231760157048232762, 6.20830695344423707404086769149, 7.09547112697851912332151090138, 8.249119455653192645744076477598, 9.138265308090915139649259479095, 9.721459430642370501499305595259, 10.23033031976023580655049014979

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