Properties

Label 2-21e2-63.16-c1-0-21
Degree $2$
Conductor $441$
Sign $0.678 + 0.734i$
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  + (−0.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + 5-s − 1.73i·6-s − 3·8-s + (1.5 + 2.59i)9-s + (−0.5 − 0.866i)10-s + 5·11-s + (1.5 − 0.866i)12-s + (−2.5 − 4.33i)13-s + (1.5 + 0.866i)15-s + (0.500 + 0.866i)16-s + (1.5 + 2.59i)17-s + (1.5 − 2.59i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (0.250 − 0.433i)4-s + 0.447·5-s − 0.707i·6-s − 1.06·8-s + (0.5 + 0.866i)9-s + (−0.158 − 0.273i)10-s + 1.50·11-s + (0.433 − 0.249i)12-s + (−0.693 − 1.20i)13-s + (0.387 + 0.223i)15-s + (0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.353 − 0.612i)18-s + (0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64937 - 0.721660i\)
\(L(\frac12)\) \(\approx\) \(1.64937 - 0.721660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68464167078627835701625326494, −10.05889829499358720750331639581, −9.385078647897478007497369838284, −8.666517176668105822070017843953, −7.47292159610009580697029651338, −6.26463411179910811009541442903, −5.20137328974966819605129078249, −3.75520227418099950711137063037, −2.69016880952267537302195258300, −1.45886971208607556344196644717, 1.74641034693958679014655227369, 3.05696271788775605588001218213, 4.22808360698426185127864388297, 6.01572280651773631726681495099, 6.91988566287984603458181768886, 7.36975048322615706487256264248, 8.584233361275081863493018719485, 9.207429231171912215799198859884, 9.813220415498522251491742169714, 11.67270127515268801840036541869

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