Properties

Label 2-21e2-441.142-c1-0-27
Degree $2$
Conductor $441$
Sign $0.645 - 0.764i$
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  + (0.0806 − 0.205i)2-s + (1.34 + 1.09i)3-s + (1.43 + 1.32i)4-s + (−1.38 + 0.665i)5-s + (0.333 − 0.186i)6-s + (1.96 − 1.76i)7-s + (0.785 − 0.378i)8-s + (0.593 + 2.94i)9-s + (0.0252 + 0.337i)10-s + (0.0398 + 0.0499i)11-s + (0.461 + 3.34i)12-s + (1.11 − 2.83i)13-s + (−0.204 − 0.546i)14-s + (−2.58 − 0.623i)15-s + (0.277 + 3.69i)16-s + (−0.801 + 0.743i)17-s + ⋯
L(s)  = 1  + (0.0570 − 0.145i)2-s + (0.773 + 0.633i)3-s + (0.715 + 0.663i)4-s + (−0.617 + 0.297i)5-s + (0.136 − 0.0763i)6-s + (0.743 − 0.668i)7-s + (0.277 − 0.133i)8-s + (0.197 + 0.980i)9-s + (0.00799 + 0.106i)10-s + (0.0120 + 0.0150i)11-s + (0.133 + 0.966i)12-s + (0.308 − 0.786i)13-s + (−0.0547 − 0.146i)14-s + (−0.666 − 0.161i)15-s + (0.0693 + 0.924i)16-s + (−0.194 + 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.645 - 0.764i$
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: \((2,\ 441,\ (\ :1/2),\ 0.645 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88836 + 0.876965i\)
\(L(\frac12)\) \(\approx\) \(1.88836 + 0.876965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
7 \( 1 + (-1.96 + 1.76i)T \)
good2 \( 1 + (-0.0806 + 0.205i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (1.38 - 0.665i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.0398 - 0.0499i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.11 + 2.83i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.801 - 0.743i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.21 + 2.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.970 - 4.25i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (6.56 + 2.02i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.22 + 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.70 + 1.75i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.558 + 0.380i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (2.01 - 1.37i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-2.11 + 5.39i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-1.62 + 0.502i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-5.64 + 3.85i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-5.96 + 5.53i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-1.86 + 3.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.60 + 15.7i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.25 - 0.792i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-1.27 - 2.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.30 - 5.86i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (3.26 + 8.30i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.13 - 8.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.12827602825416721939295143205, −10.59846998346881473556606528153, −9.480208884293950279238176467803, −8.199198808750976211896658511715, −7.79021828635049828804835910975, −6.98726573031315032277994903168, −5.30808733409950750199345889058, −3.94417389762925154394721863005, −3.44065518294379435332989411320, −2.03282112755228068732760539550, 1.45378508492057263562988798380, 2.50994465110167044888206112150, 4.05947081089368617701933553890, 5.36557597785206414432092209934, 6.47390879964980260525601967616, 7.32214796082812081470590303510, 8.260714311763667319118904130837, 8.922334058021433698695131615309, 10.04911621444670375373110272674, 11.24921530033594441062768211946

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