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Properties

Label 2-21e2-63.58-c1-0-27
Degree $2$
Conductor $441$
Sign $0.997 + 0.0690i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.69·2-s + (1.29 − 1.15i)3-s + 0.888·4-s + (1.79 + 3.10i)5-s + (2.19 − 1.95i)6-s − 1.88·8-s + (0.349 − 2.97i)9-s + (3.04 + 5.28i)10-s + (1.40 − 2.43i)11-s + (1.15 − 1.02i)12-s + (−0.5 + 0.866i)13-s + (5.89 + 1.95i)15-s − 4.98·16-s + (2.05 + 3.56i)17-s + (0.594 − 5.06i)18-s + (0.444 − 0.769i)19-s + ⋯
L(s)  = 1  + 1.20·2-s + (0.747 − 0.664i)3-s + 0.444·4-s + (0.802 + 1.38i)5-s + (0.897 − 0.798i)6-s − 0.667·8-s + (0.116 − 0.993i)9-s + (0.964 + 1.67i)10-s + (0.423 − 0.733i)11-s + (0.332 − 0.295i)12-s + (−0.138 + 0.240i)13-s + (1.52 + 0.505i)15-s − 1.24·16-s + (0.498 + 0.863i)17-s + (0.140 − 1.19i)18-s + (0.101 − 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.997 + 0.0690i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.997 + 0.0690i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.19854 - 0.110534i\)
\(L(\frac12)\) \(\approx\) \(3.19854 - 0.110534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.29 + 1.15i)T \)
7 \( 1 \)
good2 \( 1 - 1.69T + 2T^{2} \)
5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.93 + 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.98T + 31T^{2} \)
37 \( 1 + (2.38 - 4.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 + (-0.0618 - 0.107i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.87T + 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + (-5.32 - 9.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + (-2.05 - 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.80 - 8.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.66 + 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.31092894947307960134453002765, −10.30624609547652837068248781828, −9.288874988833304743744170141396, −8.332034142082270252333057733454, −6.97792357142336185268659515014, −6.38947494506779206048918201809, −5.60566595737931378758026433268, −3.88442579329640054648194177024, −3.12636563099194181617887947327, −2.09913181216847841192600381992, 1.92044022032468868539042285289, 3.37425689220625176328824464038, 4.45355292663987074359726640677, 5.11097080909143445566771258165, 5.86251386362576078051762755759, 7.47689880836455377624597381102, 8.640364330939006334162401400967, 9.514753797231341163493521493633, 9.778938370457360887019816519325, 11.39565144664174965634405439963

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