Properties

Label 2-21e2-63.38-c1-0-6
Degree $2$
Conductor $441$
Sign $0.589 - 0.808i$
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  − 2.08i·2-s + (0.845 + 1.51i)3-s − 2.34·4-s + (−1.65 + 2.86i)5-s + (3.15 − 1.76i)6-s + 0.717i·8-s + (−1.57 + 2.55i)9-s + (5.96 + 3.44i)10-s + (−2.30 + 1.33i)11-s + (−1.98 − 3.54i)12-s + (−2.11 + 1.21i)13-s + (−5.72 − 0.0790i)15-s − 3.19·16-s + (−3.59 + 6.21i)17-s + (5.32 + 3.27i)18-s + (4.24 − 2.45i)19-s + ⋯
L(s)  = 1  − 1.47i·2-s + (0.487 + 0.872i)3-s − 1.17·4-s + (−0.738 + 1.27i)5-s + (1.28 − 0.719i)6-s + 0.253i·8-s + (−0.523 + 0.851i)9-s + (1.88 + 1.08i)10-s + (−0.694 + 0.401i)11-s + (−0.572 − 1.02i)12-s + (−0.585 + 0.338i)13-s + (−1.47 − 0.0203i)15-s − 0.798·16-s + (−0.870 + 1.50i)17-s + (1.25 + 0.771i)18-s + (0.974 − 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.589 - 0.808i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.589 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900896 + 0.458122i\)
\(L(\frac12)\) \(\approx\) \(0.900896 + 0.458122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.845 - 1.51i)T \)
7 \( 1 \)
good2 \( 1 + 2.08iT - 2T^{2} \)
5 \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.30 - 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.11 - 1.21i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.59 - 6.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 + 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.32 - 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.50 - 3.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.66iT - 31T^{2} \)
37 \( 1 + (-0.844 - 1.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.553 + 0.958i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 + (8.94 + 5.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.13T + 59T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 - 8.33T + 67T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + (-6.94 - 4.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 + (1.04 - 1.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.541 - 0.937i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.47 - 5.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98426005764434825483181260559, −10.61657911264187846713447738382, −9.825766870798371127424920307130, −8.932683829509215091757656563172, −7.75367068866513254013942639134, −6.76905842941389186233510717461, −4.94395618440392725141906126704, −3.94188944106771083348443864586, −3.10417167881754960508732599887, −2.28829820594442818077958453001, 0.58676504078377032266380598458, 2.81144384447879671086137773326, 4.63420260585152418262628834610, 5.31648666692333066466137263750, 6.52327173037381270790393922042, 7.53021752003594563420652868552, 7.945543933676816260336222737482, 8.773171379039619046491685643506, 9.436508828517757990005461877115, 11.25341843965553811653577571601

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