Properties

Label 2-21e2-441.142-c1-0-45
Degree $2$
Conductor $441$
Sign $0.617 + 0.786i$
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  + (0.0182 − 0.0465i)2-s + (0.00478 − 1.73i)3-s + (1.46 + 1.35i)4-s + (3.32 − 1.59i)5-s + (−0.0805 − 0.0318i)6-s + (1.96 − 1.76i)7-s + (0.180 − 0.0867i)8-s + (−2.99 − 0.0165i)9-s + (−0.0137 − 0.183i)10-s + (−0.0359 − 0.0450i)11-s + (2.36 − 2.52i)12-s + (−0.421 + 1.07i)13-s + (−0.0463 − 0.123i)14-s + (−2.75 − 5.76i)15-s + (0.297 + 3.97i)16-s + (−4.28 + 3.97i)17-s + ⋯
L(s)  = 1  + (0.0129 − 0.0329i)2-s + (0.00276 − 0.999i)3-s + (0.732 + 0.679i)4-s + (1.48 − 0.715i)5-s + (−0.0328 − 0.0130i)6-s + (0.743 − 0.668i)7-s + (0.0636 − 0.0306i)8-s + (−0.999 − 0.00552i)9-s + (−0.00435 − 0.0581i)10-s + (−0.0108 − 0.0135i)11-s + (0.681 − 0.730i)12-s + (−0.116 + 0.297i)13-s + (−0.0124 − 0.0331i)14-s + (−0.711 − 1.48i)15-s + (0.0744 + 0.993i)16-s + (−1.03 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.617 + 0.786i$
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: \((2,\ 441,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84982 - 0.900219i\)
\(L(\frac12)\) \(\approx\) \(1.84982 - 0.900219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.00478 + 1.73i)T \)
7 \( 1 + (-1.96 + 1.76i)T \)
good2 \( 1 + (-0.0182 + 0.0465i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-3.32 + 1.59i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.0359 + 0.0450i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.421 - 1.07i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (4.28 - 3.97i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (3.80 - 6.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.40 + 6.15i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (3.42 + 1.05i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (3.17 - 5.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.11 + 2.19i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-1.99 - 1.35i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-5.78 + 3.94i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-3.09 + 7.87i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-0.957 + 0.295i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-9.58 + 6.53i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (8.17 - 7.58i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (2.65 - 4.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.721 - 3.16i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.401 + 0.0604i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (1.71 + 2.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.73 - 12.0i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.656 + 1.67i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.04 - 5.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.91368564883405442782392328636, −10.39061445925739066030550809647, −8.750208989351528874500693587013, −8.395543271240374486661734527073, −7.17071345669128596347189066376, −6.41413265099488556055613417741, −5.52954214670722399194549928060, −4.05981796265497015037304506593, −2.18525853731779033831102702556, −1.65644027149850988999247163052, 2.10256209005174510258807949494, 2.79595404627349476891211580725, 4.81864145200843907690769456247, 5.54172156710172812360208138843, 6.29787785079429426366357140923, 7.37709179914414862995675615257, 9.199263889462594329183532773828, 9.293017589758925132563091040576, 10.52072161328493631878930513288, 11.02036297988288568391739686261

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