Properties

Label 2-21e2-1.1-c3-0-3
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

\(L(2)\) \(\approx\) \(0.3226093036\)
\(L(\frac12)\) \(\approx\) \(0.3226093036\)
\(L(\frac{5}{2})\) 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

−10.79973512495689534708078588469, −9.538454907944189425434505283048, −8.831197070511296731294948696768, −8.174103879516386550422152524176, −7.19306865015937123010609795056, −6.64374548419180245277275754085, −4.59656428643991207828284026335, −3.83967849859209629889126053619, −2.03222308249388322401498824546, −0.43543910585785297948831161221, 0.43543910585785297948831161221, 2.03222308249388322401498824546, 3.83967849859209629889126053619, 4.59656428643991207828284026335, 6.64374548419180245277275754085, 7.19306865015937123010609795056, 8.174103879516386550422152524176, 8.831197070511296731294948696768, 9.538454907944189425434505283048, 10.79973512495689534708078588469

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