Properties

Label 2-21e2-1.1-c3-0-13
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  + 3·2-s + 4-s − 18·5-s − 21·8-s − 54·10-s + 36·11-s + 34·13-s − 71·16-s + 42·17-s + 124·19-s − 18·20-s + 108·22-s + 199·25-s + 102·26-s − 102·29-s + 160·31-s − 45·32-s + 126·34-s + 398·37-s + 372·38-s + 378·40-s − 318·41-s − 268·43-s + 36·44-s + 240·47-s + 597·50-s + 34·52-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 1.60·5-s − 0.928·8-s − 1.70·10-s + 0.986·11-s + 0.725·13-s − 1.10·16-s + 0.599·17-s + 1.49·19-s − 0.201·20-s + 1.04·22-s + 1.59·25-s + 0.769·26-s − 0.653·29-s + 0.926·31-s − 0.248·32-s + 0.635·34-s + 1.76·37-s + 1.58·38-s + 1.49·40-s − 1.21·41-s − 0.950·43-s + 0.123·44-s + 0.744·47-s + 1.68·50-s + 0.0906·52-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

\(L(2)\) \(\approx\) \(2.244917358\)
\(L(\frac12)\) \(\approx\) \(2.244917358\)
\(L(\frac{5}{2})\) 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 \( 1 - 3 T + p^{3} T^{2} \)
5 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 - 34 T + p^{3} T^{2} \)
17 \( 1 - 42 T + p^{3} T^{2} \)
19 \( 1 - 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 102 T + p^{3} T^{2} \)
31 \( 1 - 160 T + p^{3} T^{2} \)
37 \( 1 - 398 T + p^{3} T^{2} \)
41 \( 1 + 318 T + p^{3} T^{2} \)
43 \( 1 + 268 T + p^{3} T^{2} \)
47 \( 1 - 240 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 + 132 T + p^{3} T^{2} \)
61 \( 1 + 398 T + p^{3} T^{2} \)
67 \( 1 - 92 T + p^{3} T^{2} \)
71 \( 1 - 720 T + p^{3} T^{2} \)
73 \( 1 - 502 T + p^{3} T^{2} \)
79 \( 1 + 1024 T + p^{3} T^{2} \)
83 \( 1 + 204 T + p^{3} T^{2} \)
89 \( 1 - 354 T + p^{3} T^{2} \)
97 \( 1 - 286 T + p^{3} T^{2} \)
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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

−11.26014131280576601338953214315, −9.764820151949806553705547112161, −8.767951993539021817750960840129, −7.889662222107988959834746006944, −6.88834055361016540511882653549, −5.78649557724850355316900052586, −4.65098036948284657545615792206, −3.79915332853922999824620412539, −3.19662142771194618453947502696, −0.841888082827454654369955675858, 0.841888082827454654369955675858, 3.19662142771194618453947502696, 3.79915332853922999824620412539, 4.65098036948284657545615792206, 5.78649557724850355316900052586, 6.88834055361016540511882653549, 7.889662222107988959834746006944, 8.767951993539021817750960840129, 9.764820151949806553705547112161, 11.26014131280576601338953214315

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