Properties

Label 2-21e2-441.142-c1-0-25
Degree $2$
Conductor $441$
Sign $0.128 + 0.991i$
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  + (0.506 − 1.29i)2-s + (−1.61 + 0.622i)3-s + (0.0568 + 0.0527i)4-s + (−3.22 + 1.55i)5-s + (−0.0153 + 2.40i)6-s + (−0.247 − 2.63i)7-s + (2.59 − 1.24i)8-s + (2.22 − 2.01i)9-s + (0.370 + 4.94i)10-s + (2.17 + 2.72i)11-s + (−0.124 − 0.0498i)12-s + (1.48 − 3.77i)13-s + (−3.52 − 1.01i)14-s + (4.24 − 4.51i)15-s + (−0.286 − 3.82i)16-s + (3.20 − 2.97i)17-s + ⋯
L(s)  = 1  + (0.358 − 0.912i)2-s + (−0.933 + 0.359i)3-s + (0.0284 + 0.0263i)4-s + (−1.44 + 0.694i)5-s + (−0.00624 + 0.980i)6-s + (−0.0936 − 0.995i)7-s + (0.917 − 0.441i)8-s + (0.741 − 0.670i)9-s + (0.117 + 1.56i)10-s + (0.656 + 0.823i)11-s + (−0.0359 − 0.0143i)12-s + (0.411 − 1.04i)13-s + (−0.942 − 0.271i)14-s + (1.09 − 1.16i)15-s + (−0.0717 − 0.957i)16-s + (0.776 − 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.128 + 0.991i$
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: \((2,\ 441,\ (\ :1/2),\ 0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832433 - 0.731780i\)
\(L(\frac12)\) \(\approx\) \(0.832433 - 0.731780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.622i)T \)
7 \( 1 + (0.247 + 2.63i)T \)
good2 \( 1 + (-0.506 + 1.29i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (3.22 - 1.55i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.17 - 2.72i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.48 + 3.77i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.20 + 2.97i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.138 - 0.240i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.652 + 2.86i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.82 + 1.79i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.63 - 1.12i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-7.36 - 5.02i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-5.99 + 4.08i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (0.963 - 2.45i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (9.47 - 2.92i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.71 + 1.16i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (8.47 - 7.86i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-4.49 + 7.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.68 - 11.7i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.877 - 0.132i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (4.01 + 6.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 - 4.67i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (4.14 + 10.5i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-6.74 + 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19806347982539105773553893223, −10.39345678277211595178704098340, −9.694303990116702390026373992684, −7.62609355849638910435728046262, −7.44898772650526324534059039532, −6.25678001002284783249666468219, −4.52170764019114834279548821872, −4.00539633931830504319004624098, −3.08128989355332255712998866239, −0.821071685804712059001853879918, 1.40571719732848129259536905397, 3.82064751982817101916948126278, 4.85059055379087366363575155533, 5.78208422180034109376133410028, 6.48622101160868289009416183859, 7.53051844814972411406485899353, 8.279045497306738599728769422650, 9.238430323343197589532005578345, 10.92819015015852158070239628961, 11.41590207924910532722796255127

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