Properties

Label 2-21e2-441.142-c1-0-29
Degree $2$
Conductor $441$
Sign $0.458 + 0.888i$
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  + (0.726 − 1.85i)2-s + (−1.21 + 1.23i)3-s + (−1.43 − 1.32i)4-s + (3.50 − 1.68i)5-s + (1.39 + 3.14i)6-s + (−0.412 + 2.61i)7-s + (0.0841 − 0.0405i)8-s + (−0.0385 − 2.99i)9-s + (−0.577 − 7.71i)10-s + (2.33 + 2.92i)11-s + (3.37 − 0.148i)12-s + (−1.94 + 4.96i)13-s + (4.53 + 2.66i)14-s + (−2.18 + 6.37i)15-s + (−0.305 − 4.08i)16-s + (3.80 − 3.53i)17-s + ⋯
L(s)  = 1  + (0.513 − 1.30i)2-s + (−0.702 + 0.711i)3-s + (−0.715 − 0.664i)4-s + (1.56 − 0.754i)5-s + (0.570 + 1.28i)6-s + (−0.156 + 0.987i)7-s + (0.0297 − 0.0143i)8-s + (−0.0128 − 0.999i)9-s + (−0.182 − 2.43i)10-s + (0.703 + 0.882i)11-s + (0.975 − 0.0427i)12-s + (−0.540 + 1.37i)13-s + (1.21 + 0.711i)14-s + (−0.563 + 1.64i)15-s + (−0.0764 − 1.02i)16-s + (0.922 − 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.458 + 0.888i$
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: \((2,\ 441,\ (\ :1/2),\ 0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64555 - 1.00306i\)
\(L(\frac12)\) \(\approx\) \(1.64555 - 1.00306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.21 - 1.23i)T \)
7 \( 1 + (0.412 - 2.61i)T \)
good2 \( 1 + (-0.726 + 1.85i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-3.50 + 1.68i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.33 - 2.92i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.94 - 4.96i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.80 + 3.53i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.337 - 0.583i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.10 + 4.85i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.40 + 2.59i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-0.159 + 0.275i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.32 + 0.717i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-3.35 - 2.28i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-1.07 + 0.736i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.160 + 0.409i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (4.50 - 1.38i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (2.78 - 1.90i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (1.34 - 1.24i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (7.75 - 13.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.577 - 2.52i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-13.3 - 2.01i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (1.44 + 2.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.980 - 2.49i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (2.85 + 7.27i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (4.95 - 8.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10271030006646089814177092482, −9.935829675749913563731804257677, −9.518205636687003522374939904440, −9.094676243924568596423010157959, −6.87256340768836728066889978197, −5.77373333008873801820109375716, −4.98650756572227871913731394371, −4.20662858936695701515860544895, −2.55600016682422331929478607694, −1.55770823226438602841535560909, 1.53589954248184087395945176502, 3.41469334159663058247029585793, 5.23832183930212575236048516848, 5.80901810131007451617514412173, 6.43114971983442470066000106364, 7.28467506768190434594205256865, 7.955505951958707334556383992790, 9.529102109579344308666645194529, 10.60683783092341258375508554159, 10.95297606022682326286179020879

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