Properties

Label 2-21e2-1.1-c3-0-34
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 7.82·4-s − 0.100·5-s − 6.55·8-s − 0.0416·10-s + 43.9·11-s + 16.6·13-s + 59.9·16-s − 121.·17-s + 127.·19-s + 0.786·20-s + 18.2·22-s − 53.5·23-s − 124.·25-s + 6.89·26-s − 235.·29-s + 18.7·31-s + 77.2·32-s − 50.3·34-s − 191.·37-s + 52.6·38-s + 0.658·40-s − 319.·41-s − 218.·43-s − 343.·44-s − 22.2·46-s + 401.·47-s + ⋯
L(s)  = 1  + 0.146·2-s − 0.978·4-s − 0.00898·5-s − 0.289·8-s − 0.00131·10-s + 1.20·11-s + 0.355·13-s + 0.936·16-s − 1.73·17-s + 1.53·19-s + 0.00879·20-s + 0.176·22-s − 0.485·23-s − 0.999·25-s + 0.0520·26-s − 1.50·29-s + 0.108·31-s + 0.426·32-s − 0.254·34-s − 0.852·37-s + 0.224·38-s + 0.00260·40-s − 1.21·41-s − 0.775·43-s − 1.17·44-s − 0.0711·46-s + 1.24·47-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: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.414T + 8T^{2} \)
5 \( 1 + 0.100T + 125T^{2} \)
11 \( 1 - 43.9T + 1.33e3T^{2} \)
13 \( 1 - 16.6T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 + 53.5T + 1.21e4T^{2} \)
29 \( 1 + 235.T + 2.43e4T^{2} \)
31 \( 1 - 18.7T + 2.97e4T^{2} \)
37 \( 1 + 191.T + 5.06e4T^{2} \)
41 \( 1 + 319.T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 - 401.T + 1.03e5T^{2} \)
53 \( 1 + 643.T + 1.48e5T^{2} \)
59 \( 1 + 11.6T + 2.05e5T^{2} \)
61 \( 1 - 12.2T + 2.26e5T^{2} \)
67 \( 1 - 669.T + 3.00e5T^{2} \)
71 \( 1 + 822.T + 3.57e5T^{2} \)
73 \( 1 + 515.T + 3.89e5T^{2} \)
79 \( 1 + 805.T + 4.93e5T^{2} \)
83 \( 1 + 394.T + 5.71e5T^{2} \)
89 \( 1 - 673.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−10.05657068243529147717160229682, −9.245276754124986470721385124011, −8.687597444370195160304228180508, −7.51279839970678956632749971633, −6.39273802010921929667185561391, −5.37504269197936174321461012309, −4.25862134111062709173977883295, −3.47535521764679561694509680839, −1.60954362499798667457814268582, 0, 1.60954362499798667457814268582, 3.47535521764679561694509680839, 4.25862134111062709173977883295, 5.37504269197936174321461012309, 6.39273802010921929667185561391, 7.51279839970678956632749971633, 8.687597444370195160304228180508, 9.245276754124986470721385124011, 10.05657068243529147717160229682

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