Properties

Label 2-21e2-441.142-c1-0-28
Degree $2$
Conductor $441$
Sign $0.901 - 0.432i$
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  + (−0.700 + 1.78i)2-s + (−1.50 + 0.851i)3-s + (−1.22 − 1.13i)4-s + (2.53 − 1.22i)5-s + (−0.462 − 3.28i)6-s + (−2.05 − 1.66i)7-s + (−0.560 + 0.269i)8-s + (1.55 − 2.56i)9-s + (0.403 + 5.38i)10-s + (0.0782 + 0.0980i)11-s + (2.82 + 0.673i)12-s + (1.84 − 4.69i)13-s + (4.40 − 2.51i)14-s + (−2.78 + 4.00i)15-s + (−0.339 − 4.53i)16-s + (3.92 − 3.64i)17-s + ⋯
L(s)  = 1  + (−0.495 + 1.26i)2-s + (−0.870 + 0.491i)3-s + (−0.614 − 0.569i)4-s + (1.13 − 0.546i)5-s + (−0.189 − 1.34i)6-s + (−0.778 − 0.627i)7-s + (−0.198 + 0.0954i)8-s + (0.516 − 0.856i)9-s + (0.127 + 1.70i)10-s + (0.0235 + 0.0295i)11-s + (0.814 + 0.194i)12-s + (0.510 − 1.30i)13-s + (1.17 − 0.671i)14-s + (−0.719 + 1.03i)15-s + (−0.0848 − 1.13i)16-s + (0.952 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.901 - 0.432i$
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: \((2,\ 441,\ (\ :1/2),\ 0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792554 + 0.180298i\)
\(L(\frac12)\) \(\approx\) \(0.792554 + 0.180298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.50 - 0.851i)T \)
7 \( 1 + (2.05 + 1.66i)T \)
good2 \( 1 + (0.700 - 1.78i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.53 + 1.22i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.0782 - 0.0980i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.84 + 4.69i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.92 + 3.64i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.421 - 0.729i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.52 + 6.68i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.484 - 0.149i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (5.01 - 8.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.98 + 1.53i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-3.97 - 2.71i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (1.27 - 0.868i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-3.10 + 7.89i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-0.276 + 0.0853i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-4.55 + 3.10i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-2.24 + 2.08i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-6.95 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.117 - 0.514i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (7.59 + 1.14i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-0.119 - 0.207i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.530 - 1.35i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.315 + 0.802i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (8.79 - 15.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76249079538466154923297017345, −10.08453146918972765167090857830, −9.391549343812806990614721682733, −8.513787646823269920413898593976, −7.23191538397398746996082952548, −6.45838298101129606849755313215, −5.62428834674724509412488704707, −5.05207590739826489433629110740, −3.29098743397248273615909149866, −0.71050304502654513886679871305, 1.50847981079479859429332901891, 2.38308482303164363724402927097, 3.77610357251576257699536619850, 5.74667116107212846484275085468, 6.11720819196463101671667020551, 7.18284030343704235278516976810, 8.803739068099451547290573973771, 9.670975440110463911356907978285, 10.15796766882918173310762376788, 11.15360229942199068010194625147

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