Properties

Label 2-21e2-441.142-c1-0-42
Degree $2$
Conductor $441$
Sign $0.673 + 0.739i$
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  + (0.276 − 0.705i)2-s + (1.34 − 1.09i)3-s + (1.04 + 0.969i)4-s + (1.69 − 0.816i)5-s + (−0.400 − 1.24i)6-s + (0.217 + 2.63i)7-s + (2.33 − 1.12i)8-s + (0.602 − 2.93i)9-s + (−0.106 − 1.42i)10-s + (−2.98 − 3.73i)11-s + (2.46 + 0.157i)12-s + (−2.16 + 5.50i)13-s + (1.92 + 0.576i)14-s + (1.38 − 2.95i)15-s + (0.0660 + 0.881i)16-s + (−1.74 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.195 − 0.498i)2-s + (0.774 − 0.632i)3-s + (0.522 + 0.484i)4-s + (0.758 − 0.365i)5-s + (−0.163 − 0.510i)6-s + (0.0822 + 0.996i)7-s + (0.826 − 0.398i)8-s + (0.200 − 0.979i)9-s + (−0.0337 − 0.449i)10-s + (−0.898 − 1.12i)11-s + (0.711 + 0.0453i)12-s + (−0.599 + 1.52i)13-s + (0.513 + 0.154i)14-s + (0.356 − 0.762i)15-s + (0.0165 + 0.220i)16-s + (−0.422 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21519 - 0.978427i\)
\(L(\frac12)\) \(\approx\) \(2.21519 - 0.978427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 + 1.09i)T \)
7 \( 1 + (-0.217 - 2.63i)T \)
good2 \( 1 + (-0.276 + 0.705i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-1.69 + 0.816i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.98 + 3.73i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.16 - 5.50i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (1.74 - 1.61i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.93 + 3.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.01 - 4.43i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.66 + 0.513i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-4.63 + 8.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.97 + 2.76i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (7.34 + 5.00i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (3.53 - 2.40i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (0.214 - 0.546i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-11.2 + 3.47i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.827 - 0.564i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-3.82 + 3.55i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (3.65 - 6.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.543 - 2.38i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.22 - 1.08i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.517 + 1.31i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.792 + 2.02i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-1.86 + 3.23i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.45228637624894169069837640356, −9.982071698808448835494971492352, −9.087554804579482556424014682721, −8.400690974835317288358959963510, −7.36293493637113136366849827485, −6.39579065297241055307674905787, −5.29094806708499601108204061163, −3.70292220477832819898308681702, −2.49222759832455225256419929278, −1.87510381518605493803912626524, 1.97861890255173175566751505990, 3.07407916997439827970444720460, 4.73403860288124103141060107092, 5.31880470173134280206450509674, 6.73815391725143453360777811812, 7.50262388819302849177556666989, 8.270856501693122649664219155588, 9.928023701593498430411523390535, 10.30087235988661321229135016683, 10.55694579248163569617160847492

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