Properties

Label 2-21e2-441.142-c1-0-38
Degree $2$
Conductor $441$
Sign $0.918 + 0.395i$
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  + (0.450 − 1.14i)2-s + (1.11 + 1.32i)3-s + (0.353 + 0.327i)4-s + (2.77 − 1.33i)5-s + (2.02 − 0.678i)6-s + (−2.54 + 0.735i)7-s + (2.75 − 1.32i)8-s + (−0.523 + 2.95i)9-s + (−0.283 − 3.78i)10-s + (−1.98 − 2.48i)11-s + (−0.0419 + 0.833i)12-s + (1.46 − 3.73i)13-s + (−0.300 + 3.24i)14-s + (4.85 + 2.19i)15-s + (−0.209 − 2.79i)16-s + (2.24 − 2.08i)17-s + ⋯
L(s)  = 1  + (0.318 − 0.811i)2-s + (0.642 + 0.766i)3-s + (0.176 + 0.163i)4-s + (1.23 − 0.597i)5-s + (0.826 − 0.277i)6-s + (−0.960 + 0.277i)7-s + (0.974 − 0.469i)8-s + (−0.174 + 0.984i)9-s + (−0.0896 − 1.19i)10-s + (−0.598 − 0.749i)11-s + (−0.0121 + 0.240i)12-s + (0.406 − 1.03i)13-s + (−0.0803 + 0.867i)14-s + (1.25 + 0.566i)15-s + (−0.0523 − 0.699i)16-s + (0.543 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.918 + 0.395i$
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: \((2,\ 441,\ (\ :1/2),\ 0.918 + 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38694 - 0.491637i\)
\(L(\frac12)\) \(\approx\) \(2.38694 - 0.491637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (2.54 - 0.735i)T \)
good2 \( 1 + (-0.450 + 1.14i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.77 + 1.33i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (1.98 + 2.48i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.46 + 3.73i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-2.24 + 2.08i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (3.97 - 6.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.53 - 6.71i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.40 - 0.740i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (0.485 - 0.840i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.97 + 2.76i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (5.30 + 3.61i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-8.41 + 5.73i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.883 + 2.25i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (6.37 - 1.96i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (7.45 - 5.08i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (6.83 - 6.34i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-5.24 + 9.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0840 - 0.368i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-4.94 - 0.746i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (7.56 + 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.82 - 4.65i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.732 - 1.86i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.05 - 8.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.62199602814528159745147979581, −10.35327395610011680818954180321, −9.508940583221490551539807329587, −8.599294199043584862923988794565, −7.61571935484450375858830350182, −5.92410633031846792218603005235, −5.31902107216076921502800307974, −3.70366663595201052146109740097, −3.04243201429285591710840089766, −1.84432485737125700725741612888, 1.87178731736621381071994713451, 2.77000290549026471837783957558, 4.52957796714953053493771959883, 6.02494239893166347606057032616, 6.62226364763476628735403797863, 6.98555697504031378508767471174, 8.261752601171593936170445836221, 9.373290328149078740150300364878, 10.17064974451958014478263811244, 10.97542716245638056309660567678

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