Properties

Label 2-21e2-147.101-c1-0-3
Degree $2$
Conductor $441$
Sign $0.749 + 0.661i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.07 − 0.815i)2-s + (2.18 + 2.03i)4-s + (−2.05 − 1.40i)5-s + (−1.91 + 1.82i)7-s + (−0.956 − 1.98i)8-s + (3.13 + 4.59i)10-s + (0.173 + 1.15i)11-s + (1.20 − 0.960i)13-s + (5.46 − 2.23i)14-s + (−0.0784 − 1.04i)16-s + (2.49 + 0.769i)17-s + (0.129 − 0.0748i)19-s + (−1.65 − 7.26i)20-s + (0.578 − 2.53i)22-s + (0.369 + 1.19i)23-s + ⋯
L(s)  = 1  + (−1.46 − 0.576i)2-s + (1.09 + 1.01i)4-s + (−0.921 − 0.627i)5-s + (−0.722 + 0.690i)7-s + (−0.338 − 0.702i)8-s + (0.991 + 1.45i)10-s + (0.0523 + 0.347i)11-s + (0.333 − 0.266i)13-s + (1.46 − 0.598i)14-s + (−0.0196 − 0.261i)16-s + (0.604 + 0.186i)17-s + (0.0297 − 0.0171i)19-s + (−0.370 − 1.62i)20-s + (0.123 − 0.540i)22-s + (0.0769 + 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458692 - 0.173405i\)
\(L(\frac12)\) \(\approx\) \(0.458692 - 0.173405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.91 - 1.82i)T \)
good2 \( 1 + (2.07 + 0.815i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (2.05 + 1.40i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.173 - 1.15i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.20 + 0.960i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-2.49 - 0.769i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.129 + 0.0748i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.369 - 1.19i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-0.805 + 0.183i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-8.87 - 5.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.79 + 8.16i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-6.87 + 3.31i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.17 + 4.41i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.43 + 3.65i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (2.28 - 2.46i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-8.87 + 6.04i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-2.83 - 3.05i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (4.19 - 7.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-14.2 - 3.26i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-10.1 + 3.98i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (0.550 + 0.953i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.71 - 4.65i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-8.96 - 1.35i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 9.69iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86144654039349746618121854258, −9.959977029244696645320578429157, −9.237621228059128408513420235890, −8.404331615669087561675500800689, −7.84465645355799725942531185638, −6.68388750086077752037696317447, −5.27963279577801653379525437961, −3.78027979360064180636951250929, −2.49525517913998938035599217292, −0.799141924870461470592816282089, 0.832446202851523863012813797311, 3.06620437723584660631861957611, 4.25389564060789290065372438571, 6.21371298316518577114911397846, 6.78650075044241075505277248399, 7.78356246078282059228978483984, 8.225626413104905920649886771633, 9.531944775307463455013977533570, 10.03213387946464584926918108779, 11.03323592294347469599710618239

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